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Permutations and Combinations
------=_NextPart_01C957DF.0FE2DFE0 Content-Location: file:///C:/49764A33/6PermutationsandCombinations_files/slide0072.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Permutations and Combinations
A k-permutation is an ordered lineup of k objects.
Permutations and Combinations
------=_NextPart_01C957DF.0FE2DFE0 Content-Location: file:///C:/49764A33/6PermutationsandCombinations_files/slide0073.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Permutations and Combinations
A k-permutation is an ordered lineup of k objects.
Example A. List a= ll the 2-permutations taken from {a, b, c}.
Permutations and Combinations
------=_NextPart_01C957DF.0FE2DFE0 Content-Location: file:///C:/49764A33/6PermutationsandCombinations_files/slide0074.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Permutations and Combinations
A k-permutation is an ordered lineup of k objects.
Example A. List a= ll the 2-permutations taken from {a, b, c}.
ab, ba, ac, ca, b= c, cb
Permutations and Combinations
------=_NextPart_01C957DF.0FE2DFE0 Content-Location: file:///C:/49764A33/6PermutationsandCombinations_files/slide0075.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Permutations and Combinations
A k-permutation is an ordered lineup of k objects.
Example A. List a= ll the 2-permutations taken from {a, b, c}.
ab, ba, ac, ca, b= c, cb
Example B. List a= ll the 3-permutations taken from {a, b, c}.
Permutations and Combinations
------=_NextPart_01C957DF.0FE2DFE0 Content-Location: file:///C:/49764A33/6PermutationsandCombinations_files/slide0076.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Permutations and Combinations
A k-permutation is an ordered lineup of k objects.
Example A. List a= ll the 2-permutations taken from {a, b, c}.
ab, ba, ac, ca, b= c, cb
Example B. List a= ll the 3-permutations taken from {a, b, c}.
abc, acb, bac, bca, cab, cba
Permutations and Combinations
------=_NextPart_01C957DF.0FE2DFE0 Content-Location: file:///C:/49764A33/6PermutationsandCombinations_files/slide0077.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Permutations and Combinations
A k-permutation is an ordered lineup of k objects.
Example A. List a= ll the 2-permutations taken from {a, b, c}.
ab, ba, ac, ca, b= c, cb
Example B. List a= ll the 3-permutations taken from {a, b, c}.
abc, acb, bac, bca, cab, cba
Permutations and Combinations
The number of k-permutations (ordered arrangements) <= span lang=3DEN-US style=3D'mso-fareast-font-family:PMingLiU;mso-hansi-font-fami= ly:Arial; font-size:133%;mso-fareast-language:ZH-TW'>taken from n objects is: =
nPk =3D
n!
(n – k)!
------=_NextPart_01C957DF.0FE2DFE0 Content-Location: file:///C:/49764A33/6PermutationsandCombinations_files/slide0071.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Permutations and Combinations
A k-permutation is an ordered lineup of k objects.
Example A. List a= ll the 2-permutations taken from {a, b, c}.
ab, ba, ac, ca, b= c, cb
Example B. List a= ll the 3-permutations taken from {a, b, c}.
abc, acb, bac, bca, cab, cba
Permutations and Combinations
The number of k-permutations (ordered arrangements) <= span lang=3DEN-US style=3D'mso-fareast-font-family:PMingLiU;mso-hansi-font-fami= ly:Arial; font-size:133%;mso-fareast-language:ZH-TW'>taken from n objects is: =
nPk =3D
n!
(n – k)!
Example C. How ma= ny 2-permutations taken from {a, b, c}
are there?=
------=_NextPart_01C957DF.0FE2DFE0 Content-Location: file:///C:/49764A33/6PermutationsandCombinations_files/slide0079.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Permutations and Combinations
A k-permutation is an ordered lineup of k objects.
Example A. List a= ll the 2-permutations taken from {a, b, c}.
ab, ba, ac, ca, b= c, cb
Example B. List a= ll the 3-permutations taken from {a, b, c}.
abc, acb, bac, bca, cab, cba
Permutations and Combinations
The number of k-permutations (ordered arrangements) <= span lang=3DEN-US style=3D'mso-fareast-font-family:PMingLiU;mso-hansi-font-fami= ly:Arial; font-size:133%;mso-fareast-language:ZH-TW'>taken from n objects is: =
nPk =3D
n!
(n – k)!
Example C. How ma= ny 2-permutations taken from {a, b, c}
are there?
=
n=3D3, k=3D2,
------=_NextPart_01C957DF.0FE2DFE0 Content-Location: file:///C:/49764A33/6PermutationsandCombinations_files/slide0080.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Permutations and Combinations
A k-permutation is an ordered lineup of k objects.
Example A. List a= ll the 2-permutations taken from {a, b, c}.
ab, ba, ac, ca, b= c, cb
Example B. List a= ll the 3-permutations taken from {a, b, c}.
abc, acb, bac, bca, cab, cba
Permutations and Combinations
The number of k-permutations (ordered arrangements) <= span lang=3DEN-US style=3D'mso-fareast-font-family:PMingLiU;mso-hansi-font-fami= ly:Arial; font-size:133%;mso-fareast-language:ZH-TW'>taken from n objects is: =
nPk =3D
n!
(n – k)!
Example C. How ma= ny 2-permutations taken from {a, b, c}
are there?
=
n=3D3, k=3D2, 3P2 =3D        = ;       
3!
(3 – 2)!
------=_NextPart_01C957DF.0FE2DFE0 Content-Location: file:///C:/49764A33/6PermutationsandCombinations_files/slide0081.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Permutations and Combinations
A k-permutation is an ordered lineup of k objects.
Example A. List a= ll the 2-permutations taken from {a, b, c}.
ab, ba, ac, ca, b= c, cb
Example B. List a= ll the 3-permutations taken from {a, b, c}.
abc, acb, bac, bca, cab, cba
Permutations and Combinations
The number of k-permutations (ordered arrangements) <= span lang=3DEN-US style=3D'mso-fareast-font-family:PMingLiU;mso-hansi-font-fami= ly:Arial; font-size:133%;mso-fareast-language:ZH-TW'>taken from n objects is: =
nPk =3D
n!
(n – k)!
Example C. How ma= ny 2-permutations taken from {a, b, c}
are there?
=
n=3D3, k=3D2, 3P2 =3D        = ;        =3D        =3D 6
3!
(3 – 2)!
6
1
------=_NextPart_01C957DF.0FE2DFE0 Content-Location: file:///C:/49764A33/6PermutationsandCombinations_files/slide0078.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Permutations and Combinations
A k-permutation is an ordered lineup of k objects.
Example A. List a= ll the 2-permutations taken from {a, b, c}.
ab, ba, ac, ca, b= c, cb
Example B. List a= ll the 3-permutations taken from {a, b, c}.
abc, acb, bac, bca, cab, cba
Permutations and Combinations
The number of k-permutations (ordered arrangements) <= span lang=3DEN-US style=3D'mso-fareast-font-family:PMingLiU;mso-hansi-font-fami= ly:Arial; font-size:133%;mso-fareast-language:ZH-TW'>taken from n objects is: =
nPk =3D
n!
(n – k)!
Example C. How ma= ny 2-permutations taken from {a, b, c}
are there?
=
n=3D3, k=3D2, 3P2 =3D        = ;        =3D        =3D 6
3!
(3 – 2)!
6
1
They are {ab, ba,= ac, ca, bc, cb}.        = ;    
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Example D. How ma= ny different arrangements of 7 people from a group of 10 people in a row of 7 seats are there?
Permutations and Combinations
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Example D. How ma= ny different arrangements of 7 people from a group of 10 people in a row of 7 seats are there?
=
There are 10 peop= le so n =3D 10.
Permutations and Combinations
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Example D. How ma= ny different arrangements of 7 people from a group of 10 people in a row of 7 seats are there?
=
There are 10 peop= le so n =3D 10. We are to seat 7 of them in order so k =3D 7,=
Permutations and Combinations
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Example D. How ma= ny different arrangements of 7 people from a group of 10 people in a row of 7 seats are there?
=
There are 10 peop= le so n =3D 10. We are to seat 7 of them in order so k =3D 7,= so there are
10P7 =3D 10! / (10 = 211; 7)!
Permutations and Combinations
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Example D. How ma= ny different arrangements of 7 people from a group of 10 people in a row of 7 seats are there?
=
There are 10 peop= le so n =3D 10. We are to seat 7 of them in order so k =3D 7,= so there are
10P7 =3D 10! / (10 = 211; 7)! =3D10! / 3!
Permutations and Combinations
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Example D. How ma= ny different arrangements of 7 people from a group of 10 people in a row of 7 seats are there?
=
There are 10 peop= le so n =3D 10. We are to seat 7 of them in order so k =3D 7,= so there are
10P7 =3D 10! / (10 = 211; 7)! =3D10! / 3!
       =3D= 10 x 9 x 8 x .. X 4 =3D 604800 possibilities.
Permutations and Combinations
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Example D. How ma= ny different arrangements of 7 people from a group of 10 people in a row of 7 seats are there?
=
There are 10 peop= le so n =3D 10. We are to seat 7 of them in order so k =3D 7,= so there are
10P7 =3D 10! / (10 = 211; 7)! =3D10! / 3!
       =3D= 10 x 9 x 8 x .. X 4 =3D 604800 possibilities.
Permutations and Combinations
A k-combination is a (unordered) collection = of k objects.
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Permutations and Combinations
A k-combination is a (unordered) collection = of k objects.
Example E. List a= ll the 2-combinations from the set {a, b, c}.
Example D. How ma= ny different arrangements of 7 people from a group of 10 people in a row of 7 seats are there?
=
There are 10 peop= le so n =3D 10. We are to seat 7 of them in order so k =3D 7,= so there are
10P7 =3D 10! / (10 = 211; 7)! =3D10! / 3!
       =3D= 10 x 9 x 8 x .. X 4 =3D 604800 possibilities.
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Permutations and Combinations
A k-combination is a (unordered) collection = of k objects.
Example E. List a= ll the 2-combinations from the set {a, b, c}.
=
{a, b}, {a, c}, {b, c}
Example D. How ma= ny different arrangements of 7 people from a group of 10 people in a row of 7 seats are there?
=
There are 10 peop= le so n =3D 10. We are to seat 7 of them in order so k =3D 7,= so there are
10P7 =3D 10! / (10 = 211; 7)! =3D10! / 3!
       =3D= 10 x 9 x 8 x .. X 4 =3D 604800 possibilities.
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Permutations and Combinations
A k-combination is a (unordered) collection = of k objects.
Example E. List a= ll the 2-combinations from the set {a, b, c}.
=
{a, b}, {a, c}, {b, c}
Example F. List a= ll the 3-combination taken from {a, b, c}.
Example D. How ma= ny different arrangements of 7 people from a group of 10 people in a row of 7 seats are there?
=
There are 10 peop= le so n =3D 10. We are to seat 7 of them in order so k =3D 7,= so there are
10P7 =3D 10! / (10 = 211; 7)! =3D10! / 3!
       =3D= 10 x 9 x 8 x .. X 4 =3D 604800 possibilities.
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Permutations and Combinations
A k-combination is a (unordered) collection = of k objects.
Example E. List a= ll the 2-combinations from the set {a, b, c}.
=
{a, b}, {a, c}, {b, c}
Example F. List a= ll the 3-combination taken from {a, b, c}.
=
{a, b, c}<= span lang=3DEN-US style=3D'mso-fareast-font-family:\5B8B\4F53;mso-hansi-font-fa= mily: Arial;font-size:133%;color:#003300;mso-fareast-language:ZH-CN;mso-special-= format: lastCR;display:none'>
Example D. How ma= ny different arrangements of 7 people from a group of 10 people in a row of 7 seats are there?
=
There are 10 peop= le so n =3D 10. We are to seat 7 of them in order so k =3D 7,= so there are
10P7 =3D 10! / (10 = 211; 7)! =3D10! / 3!
       =3D= 10 x 9 x 8 x .. X 4 =3D 604800 possibilities.
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Permutations and Combinations
A k-combination is a (unordered) collection = of k objects.
Example E. List a= ll the 2-combinations from the set {a, b, c}.
=
{a, b}, {a, c}, {b, c}
Example F. List a= ll the 3-combination taken from {a, b, c}.
=
{a, b, c}<= span lang=3DEN-US style=3D'mso-fareast-font-family:\5B8B\4F53;mso-hansi-font-fa= mily: Arial;font-size:133%;color:#003300;mso-fareast-language:ZH-CN;mso-special-= format: lastCR;display:none'>
The number of k-combinations (unordered collections) = taken = from n objects is: <= /span>
nCk =3D
n!
(n – k)!k!
Example D. How ma= ny different arrangements of 7 people from a group of 10 people in a row of 7 seats are there?
=
There are 10 peop= le so n =3D 10. We are to seat 7 of them in order so k =3D 7,= so there are
10P7 =3D 10! / (10 = 211; 7)! =3D10! / 3!
       =3D= 10 x 9 x 8 x .. X 4 =3D 604800 possibilities.
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Example G. How many 2-combinations from the set {a, b, c} are there?
Permutations and Combinations
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Example G. How many 2-combinations from the set {a, b, c}<= span lang=3DEN-US style=3D'mso-fareast-font-family:PMingLiU;mso-hansi-font-fami= ly:Arial; font-size:133%;color:#003300;mso-fareast-language:ZH-CN'> are there?
=
n =3D 3, k =3D 2,=
Permutations and Combinations
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Example G. How many 2-combinations from the set {a, b, c}<= span lang=3DEN-US style=3D'mso-fareast-font-family:PMingLiU;mso-hansi-font-fami= ly:Arial; font-size:133%;color:#003300;mso-fareast-language:ZH-CN'> are there?
=
n =3D 3, k =3D 2,= hence there are
=
3C2 =3D        = ;         
3!
(3 – <= span lang=3DEN-US style=3D'mso-fareast-font-family:PMingLiU;mso-hansi-font-fami= ly:Arial; font-size:133%;color:#003300;mso-fareast-language:ZH-CN'>2)!2!
Permutations and Combinations
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Example G. How many 2-combinations from the set {a, b, c}<= span lang=3DEN-US style=3D'mso-fareast-font-family:PMingLiU;mso-hansi-font-fami= ly:Arial; font-size:133%;color:#003300;mso-fareast-language:ZH-CN'> are there?
=
n =3D 3, k =3D 2,= hence there are
=
3C2 =3D        = ;         
3!
(3 – <= span lang=3DEN-US style=3D'mso-fareast-font-family:PMingLiU;mso-hansi-font-fami= ly:Arial; font-size:133%;color:#003300;mso-fareast-language:ZH-CN'>2)!2!
Permutations and Combinations
3
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Example G. How many 2-combinations from the set {a, b, c}<= span lang=3DEN-US style=3D'mso-fareast-font-family:PMingLiU;mso-hansi-font-fami= ly:Arial; font-size:133%;color:#003300;mso-fareast-language:ZH-CN'> are there?
=
n =3D 3, k =3D 2,= hence there are
=
3C2 =3D        = ;          =3D 3 
3!
(3 – <= span lang=3DEN-US style=3D'mso-fareast-font-family:PMingLiU;mso-hansi-font-fami= ly:Arial; font-size:133%;color:#003300;mso-fareast-language:ZH-CN'>2)!2!
Permutations and Combinations
3
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Example G. How many 2-combinations from the set {a, b, c}<= span lang=3DEN-US style=3D'mso-fareast-font-family:PMingLiU;mso-hansi-font-fami= ly:Arial; font-size:133%;color:#003300;mso-fareast-language:ZH-CN'> are there?
=
n =3D 3, k =3D 2,= hence there are
=
3C2 =3D        = ;          =3D 3 
=
So there are three 2-combinations.
3!
(3 – <= span lang=3DEN-US style=3D'mso-fareast-font-family:PMingLiU;mso-hansi-font-fami= ly:Arial; font-size:133%;color:#003300;mso-fareast-language:ZH-CN'>2)!2!
Permutations and Combinations
3
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Example G. How many 2-combinations from the set {a, b, c}<= span lang=3DEN-US style=3D'mso-fareast-font-family:PMingLiU;mso-hansi-font-fami= ly:Arial; font-size:133%;color:#003300;mso-fareast-language:ZH-CN'> are there?
=
n =3D 3, k =3D 2,= hence there are
=
3C2 =3D        = ;          =3D 3 
=
So there are three 2-combinations.
3!
(3 – <= span lang=3DEN-US style=3D'mso-fareast-font-family:PMingLiU;mso-hansi-font-fami= ly:Arial; font-size:133%;color:#003300;mso-fareast-language:ZH-CN'>2)!2!
Permutations and Combinations
They are {<= /span>a, b}, {a, c}, {b, c}.
3
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Example G. How many 2-combinations from the set {a, b, c}<= span lang=3DEN-US style=3D'mso-fareast-font-family:PMingLiU;mso-hansi-font-fami= ly:Arial; font-size:133%;color:#003300;mso-fareast-language:ZH-CN'> are there?
=
n =3D 3, k =3D 2,= hence there are
=
3C2 =3D        = ;          =3D 3 
=
So there are three 2-combinations.
3!
(3 – <= span lang=3DEN-US style=3D'mso-fareast-font-family:PMingLiU;mso-hansi-font-fami= ly:Arial; font-size:133%;color:#003300;mso-fareast-language:ZH-CN'>2)!2!
Permutations and Combinations
They are {<= /span>a, b}, {a, c}, {b, c}.
Example H. A  Chinese take-out offers a beef dish, a chicken dish, a vegetable dish, fried rice, and fried noodles.  We may chose any 3 of them for a 3-Combo Special. How many different 3-Combo Specials are possible?
3
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Example G. How many 2-combinations from the set {a, b, c}<= span lang=3DEN-US style=3D'mso-fareast-font-family:PMingLiU;mso-hansi-font-fami= ly:Arial; font-size:133%;color:#003300;mso-fareast-language:ZH-CN'> are there?
=
n =3D 3, k =3D 2,= hence there are
=
3C2 =3D        = ;          =3D 3 
=
So there are three 2-combinations.
3!
(3 – <= span lang=3DEN-US style=3D'mso-fareast-font-family:PMingLiU;mso-hansi-font-fami= ly:Arial; font-size:133%;color:#003300;mso-fareast-language:ZH-CN'>2)!2!
Permutations and Combinations
They are {<= /span>a, b}, {a, c}, {b, c}.
Example H. A  Chinese take-out offers a beef dish, a chicken dish, a vegetable dish, fried rice, and fried noodles.  We may chose any 3 of them for a 3-Combo Special. How many different 3-Combo Specials are possible? <= /span>
   
n =3D 5, we are t= o take 3 of so k =3D 3,
3
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Example G. How many 2-combinations from the set {a, b, c}<= span lang=3DEN-US style=3D'mso-fareast-font-family:PMingLiU;mso-hansi-font-fami= ly:Arial; font-size:133%;color:#003300;mso-fareast-language:ZH-CN'> are there?
=
n =3D 3, k =3D 2,= hence there are
=
3C2 =3D        = ;          =3D 3 
=
So there are three 2-combinations.
3!
(3 – <= span lang=3DEN-US style=3D'mso-fareast-font-family:PMingLiU;mso-hansi-font-fami= ly:Arial; font-size:133%;color:#003300;mso-fareast-language:ZH-CN'>2)!2!
Permutations and Combinations
They are {<= /span>a, b}, {a, c}, {b, c}.
Example H. A  Chinese take-out offers a beef dish, a chicken dish, a vegetable dish, fried rice, and fried noodles.  We may chose any 3 of them for a 3-Combo Special. How many different 3-Combo Specials are possible? <= /span>
   
n =3D 5, we are t= o take 3 of so k =3D 3, hence there are
5!
(5 – 3)!3!
5C3 =3D        = ;            &n= bsp;  
3
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Example G. How many 2-combinations from the set {a, b, c}<= span lang=3DEN-US style=3D'mso-fareast-font-family:PMingLiU;mso-hansi-font-fami= ly:Arial; font-size:133%;color:#003300;mso-fareast-language:ZH-CN'> are there?
=
n =3D 3, k =3D 2,= hence there are
=
3C2 =3D        = ;          =3D 3 
=
So there are three 2-combinations.
3!
(3 – <= span lang=3DEN-US style=3D'mso-fareast-font-family:PMingLiU;mso-hansi-font-fami= ly:Arial; font-size:133%;color:#003300;mso-fareast-language:ZH-CN'>2)!2!
Permutations and Combinations
They are {<= /span>a, b}, {a, c}, {b, c}.
Example H. A  Chinese take-out offers a beef dish, a chicken dish, a vegetable dish, fried rice, and fried noodles.  We may chose any 3 of them for a 3-Combo Special. How many different 3-Combo Specials are possible? <= /span>
   
n =3D 5, we are t= o take 3 of so k =3D 3, hence there are
5!
(5 – 3)!3!
5C3 =3D        = ;          =3D        
5!
2!3!
3
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Example G. How many 2-combinations from the set {a, b, c}<= span lang=3DEN-US style=3D'mso-fareast-font-family:PMingLiU;mso-hansi-font-fami= ly:Arial; font-size:133%;color:#003300;mso-fareast-language:ZH-CN'> are there?
=
n =3D 3, k =3D 2,= hence there are
=
3C2 =3D        = ;          =3D 3 
=
So there are three 2-combinations.
3!
(3 – <= span lang=3DEN-US style=3D'mso-fareast-font-family:PMingLiU;mso-hansi-font-fami= ly:Arial; font-size:133%;color:#003300;mso-fareast-language:ZH-CN'>2)!2!
Permutations and Combinations
They are {<= /span>a, b}, {a, c}, {b, c}.
Example H. A  Chinese take-out offers a beef dish, a chicken dish, a vegetable dish, fried rice, and fried noodles.  We may chose any 3 of them for a 3-Combo Special. How many different 3-Combo Specials are possible? <= /span>
   
n =3D 5, we are t= o take 3 of so k =3D 3, hence there are
5!
(5 – 3)!3!
5C3 =3D        = ;          =3D        
5!
2!3!
3
5*4
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Example G. How many 2-combinations from the set {a, b, c}<= span lang=3DEN-US style=3D'mso-fareast-font-family:PMingLiU;mso-hansi-font-fami= ly:Arial; font-size:133%;color:#003300;mso-fareast-language:ZH-CN'> are there?
=
n =3D 3, k =3D 2,= hence there are
=
3C2 =3D        = ;          =3D 3 
=
So there are three 2-combinations.
3!
(3 – <= span lang=3DEN-US style=3D'mso-fareast-font-family:PMingLiU;mso-hansi-font-fami= ly:Arial; font-size:133%;color:#003300;mso-fareast-language:ZH-CN'>2)!2!
Permutations and Combinations
They are {<= /span>a, b}, {a, c}, {b, c}.
Example H. A  Chinese take-out offers a beef dish, a chicken dish, a vegetable dish, fried rice, and fried noodles.  We may chose any 3 of them for a 3-Combo Special. How many different 3-Combo Specials are possible? <= /span>
   
n =3D 5, we are t= o take 3 of so k =3D 3, hence there are
5!
(5 – 3)!3!
5C3 =3D        = ;          =3D         =3D 10  3-Combos speci= als.&#= 13;
5!
2!3!
3
5*4
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