- #36

anuttarasammyak

Gold Member

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- 1,296

Following up the post ##32 and #33 I show here counting relevant cases of N(10), N(11), N(12), N(13) and N(14). N(n) is introduced in post #32.

N(10)

##Three\ 3-of-a-kind##

For brevity I introduce < > notation for number of multiset multiset permutation of the case AAABBBCCCD

[tex] <3+3+3+1> = \frac{10!}{3!3!1!}[/tex]

##<3+3+3+1>*5*4 ##... choice D,E

##Two\ 3-of-a-kind##

AAABBBCCCC

##<3+3+4>*5*\ 4C_2## ...choice C,(AB)

AAABBBCCDD

##<3+3+2+2>*5*\ 4C_2## ...choice E,(AB)

AAABBBCCDE

##<3+3+2+1+1>*5*\ 4C_2## ...choice C,(AB)

##One\ 3-of-a-kind##

AAABBBBBBB (*)

##<3+7>*5*4## ...choice A,B

AAABBBBBBC

##<3+6+1>*5*4*3## ...choice A,B,C

AAABBBBBCC

##<3+5+2>*5*4*3##... choice A,B,C

AAABBBBBCD

##<3+5+1+1>*5*4*3## ...choice A,B,E

AAABBBBCCD

##<3+4+2+1>*5*4*3*2##... choice A,B,C,D

AAABBBBCDE

##<3+4+1+1+1>*5*4##... choice A,B

AAABBBBCDE

##<3+2+2+2+1>*5*4##... choice A,B

##7-of-a-kind##

In addition to (*)

AAAAAAABBC

##<7+2+1>*5*4*3## ...choice A,B,C

AAAAAAABCD

##<7+1+1+1>*5*4## ...choice A,E********************************************

N(11)

##Two\ 4-of-a-kind##

AAAABBBBCCC

##<4+4+3>*5*\ _4C_2 ##... choice C,(AB)

AAAABBBBCCD

##<4+4+2+1>*5*4*3##... choice C,D,E

AAAABBBBCDE

##<4+4+1+1+1>*\ _5C_2##... choice (AB)

##One\ 4-of-a-kind##

AAAABBBBBBB (*)

##<4+7>*5*4## choice A,B

AAAABBBBBBC

##<4+6+1>*5*4*3## choice A,B,C

AAAABBBBBCC

##<4+5+2>*5*4*3## choice A,B,C

AAAABBBBBCD

##<4+5+1+1>*5*4*3## choice A,B,E

AAABBBCCCD

##<4+3+3+1>*5*4*3## choice A,D,E

AAAABBBCCDD

##<4+3+2+2>*5*4*3## choice A,B,E

AAAABBBCCDE

##<4+3+2+1+1>*5*4## choice A,B

##7-of-a-kind##

In addition to (*)

AAAAAAABBBC

##<7+3+1>*5*4*3## choice A,B,C

AAAAAAABBCC

##<7+2+2>*5*\ _4C_2## choice A,(BC)

AAAAAAABBCD

##<7+2+1+1>*5*4*3## choice A,B,E

AAAAAAABCDE

##<7+1+1+1+1>*5## choice A

*****************************************

N(12)

##Two\ 5-of-a-kind##

AAAAABBBBBCC

##<5+5+2>*5*\ _4C_2 ##... choice C,(AB)

AAAAABBBBBCD

##<5+5+1+1>*5*\ _4C_2 ##... choice E,(AB)

##One\ 5-of-a-kind##

AAAAABBBBBBB (*)

##<5+7>*5*4## choice A,B

AAAAABBBBBBC

##<5+6+1>*5*4*3## choice A,B,C

AAAAABBBBCCC

##<5+4+3>*5*4*3## choice A,B,C

AAAAABBBBCCD

##<5+4+2+1>*5*4*3*2## choice A,B,C,D

AAAAABBBCCCD

##<5+3+3+1>*5*4*3## choice A,D,E

AAAAABBBCCDD

##<5+3+2+2>*5*4*3## choice A,B,E

AAAAABBBBCDE

##<5+4+1+1+1>*5*4## choice A,B

AAAAABBBCCDE

##<5+3+2+1+1>*5*4*3## choice A,B,C

AAAAABBCCDDE

##<5+2+2+2+1>*5*4## choice A,E

##7-of-a-kind##

In addition to (*)

AAAAAAABBBBC

##<7+4+1>*5*4*3## choice A,B,C

AAAAAAABBBCC

##<7+3+2>*5*4## choice A,B,C

AAAAAAABBBCD

##<7+3+1+1>*5*4*3## choice A,B,E

AAAAAAABBCCD

##<7+2+2+1>*5*4*3## choice A,D,E

AAAAAAABBCDE

##<7+2+1+1+1>*5*4## choice A,B

*****************************************

N(13)

##Two\ 6-of-a-kind##

AAAAAABBBBBC

##<6+6+1>*5*\ _4C_2 ##... choice C,(AB)

##One\ 6-of-a-kind##

AAAAAABBBBBBB (*)

##<6+7>*5*4## choice A,B

AAAAAABBBBBCC

##<6+5+2>*5*4*3## choice A,B,C

AAAAAABBBBCCC

##<6+4+3>*5*4*3## choice A,B,C

AAAAAABBBBBCD

##<6+5+1+1>*5*4*3## choice A,B,E

AAAAAABBBBCCD

##<6+4+2+1>*5*4*3*2## choice A,B,C,D

AAAAAABBBCCCD

##<6+3+3+1>*5*4*3## choice A,D,E

AAAAAABBBCCDD

##<6+3+2+2>*5*4*3## choice A,B,E

##7-of-a-kind##

In addition to (*)

AAAAAAABBBBBC

##<7+5+1>*5*4*3## choice A,B,C

AAAAAAABBBBCC

##<7+4+2>*5*4*3## choice A,B,C

AAAAAAABBBCCC

##<7+3+3>*5*\ _4C_2## choice A,(BC)

AAAAAAABBBBCC

##<7+4+1+1>*5*4*3## choice A,(BC)

AAAAAAABBBCCD

##<7+3+2+1>*5*4*3*2## choice A,B,C,D

AAAAAAABBCCDD

##<7+2+2+2>*5*4## choice A,E

AAAAAAABBBCDE

##<7+3+1+1+1>*5*4## choice A,B

AAAAAAABBCCDE

##<7+2+2+1+1>*5*\ _4C_2## choice A,(BC)

****************************

N(14)

##Two\ 7-of-a-kind##

AAAAAAABBBBBBB

##<7+7>*5*4##... choice A,B

##One\ 7-of-a-kind##

AAAAAAABBBBBBC

##<7+6+1>*5*4*3## choice A,B,C

AAAAAAABBBBBCC

##<7+5+2>*5*4*3## choice A,B,C

AAAAAAABBBBCCC

##<7+4+3>*5*4*3## choice A,B,C

AAAAAAABBBBBCD

##<7+5+1+1>*5*4*3## choice A,B,E

AAAAAAABBBBCCD

##<7+4+2+1>*5*4*3*2## choice A,B,C,D

AAAAAAABBBCCCD

##<7+3+3+1>*5*4*3## choice A,D,E

AAAAAAABBBCCDD

##<7+3+2+2>*5*4*3## choice A,B,E

AAAAAAABBBBCDE

##<7+4+1+1+1>*5*4*3## choice A,B,E

AAAAAAABBBCCDE

##<7+3+2+1+1>*5*4*3## choice A,B,C

AAAAAAABBCCDDE

##<7+2+2+2+1>*5*4## choice A,E

**********************************

*************************************anuttarasammyak said:*********************************************************

Is the number of cases we want is written as

14∑n=7 14Cn∗N(n)∑n=714 14Cn∗N(n)

\sum_{n=7}^{14} \ _{14}C_n*N(n)

where N(n) is the number of cases in game of n dices of five faces,

satisfying 7-of-a-kind or (n-7)-of-a-kind with no redundant count ?

*********************************************************

N(10)

##Three\ 3-of-a-kind##

For brevity I introduce < > notation for number of multiset multiset permutation of the case AAABBBCCCD

[tex] <3+3+3+1> = \frac{10!}{3!3!1!}[/tex]

##<3+3+3+1>*5*4 ##... choice D,E

##Two\ 3-of-a-kind##

AAABBBCCCC

##<3+3+4>*5*\ 4C_2## ...choice C,(AB)

AAABBBCCDD

##<3+3+2+2>*5*\ 4C_2## ...choice E,(AB)

AAABBBCCDE

##<3+3+2+1+1>*5*\ 4C_2## ...choice C,(AB)

##One\ 3-of-a-kind##

AAABBBBBBB (*)

##<3+7>*5*4## ...choice A,B

AAABBBBBBC

##<3+6+1>*5*4*3## ...choice A,B,C

AAABBBBBCC

##<3+5+2>*5*4*3##... choice A,B,C

AAABBBBBCD

##<3+5+1+1>*5*4*3## ...choice A,B,E

AAABBBBCCD

##<3+4+2+1>*5*4*3*2##... choice A,B,C,D

AAABBBBCDE

##<3+4+1+1+1>*5*4##... choice A,B

AAABBBBCDE

##<3+2+2+2+1>*5*4##... choice A,B

##7-of-a-kind##

In addition to (*)

AAAAAAABBC

##<7+2+1>*5*4*3## ...choice A,B,C

AAAAAAABCD

##<7+1+1+1>*5*4## ...choice A,E********************************************

N(11)

##Two\ 4-of-a-kind##

AAAABBBBCCC

##<4+4+3>*5*\ _4C_2 ##... choice C,(AB)

AAAABBBBCCD

##<4+4+2+1>*5*4*3##... choice C,D,E

AAAABBBBCDE

##<4+4+1+1+1>*\ _5C_2##... choice (AB)

##One\ 4-of-a-kind##

AAAABBBBBBB (*)

##<4+7>*5*4## choice A,B

AAAABBBBBBC

##<4+6+1>*5*4*3## choice A,B,C

AAAABBBBBCC

##<4+5+2>*5*4*3## choice A,B,C

AAAABBBBBCD

##<4+5+1+1>*5*4*3## choice A,B,E

AAABBBCCCD

##<4+3+3+1>*5*4*3## choice A,D,E

AAAABBBCCDD

##<4+3+2+2>*5*4*3## choice A,B,E

AAAABBBCCDE

##<4+3+2+1+1>*5*4## choice A,B

##7-of-a-kind##

In addition to (*)

AAAAAAABBBC

##<7+3+1>*5*4*3## choice A,B,C

AAAAAAABBCC

##<7+2+2>*5*\ _4C_2## choice A,(BC)

AAAAAAABBCD

##<7+2+1+1>*5*4*3## choice A,B,E

AAAAAAABCDE

##<7+1+1+1+1>*5## choice A

*****************************************

N(12)

##Two\ 5-of-a-kind##

AAAAABBBBBCC

##<5+5+2>*5*\ _4C_2 ##... choice C,(AB)

AAAAABBBBBCD

##<5+5+1+1>*5*\ _4C_2 ##... choice E,(AB)

##One\ 5-of-a-kind##

AAAAABBBBBBB (*)

##<5+7>*5*4## choice A,B

AAAAABBBBBBC

##<5+6+1>*5*4*3## choice A,B,C

AAAAABBBBCCC

##<5+4+3>*5*4*3## choice A,B,C

AAAAABBBBCCD

##<5+4+2+1>*5*4*3*2## choice A,B,C,D

AAAAABBBCCCD

##<5+3+3+1>*5*4*3## choice A,D,E

AAAAABBBCCDD

##<5+3+2+2>*5*4*3## choice A,B,E

AAAAABBBBCDE

##<5+4+1+1+1>*5*4## choice A,B

AAAAABBBCCDE

##<5+3+2+1+1>*5*4*3## choice A,B,C

AAAAABBCCDDE

##<5+2+2+2+1>*5*4## choice A,E

##7-of-a-kind##

In addition to (*)

AAAAAAABBBBC

##<7+4+1>*5*4*3## choice A,B,C

AAAAAAABBBCC

##<7+3+2>*5*4## choice A,B,C

AAAAAAABBBCD

##<7+3+1+1>*5*4*3## choice A,B,E

AAAAAAABBCCD

##<7+2+2+1>*5*4*3## choice A,D,E

AAAAAAABBCDE

##<7+2+1+1+1>*5*4## choice A,B

*****************************************

N(13)

##Two\ 6-of-a-kind##

AAAAAABBBBBC

##<6+6+1>*5*\ _4C_2 ##... choice C,(AB)

##One\ 6-of-a-kind##

AAAAAABBBBBBB (*)

##<6+7>*5*4## choice A,B

AAAAAABBBBBCC

##<6+5+2>*5*4*3## choice A,B,C

AAAAAABBBBCCC

##<6+4+3>*5*4*3## choice A,B,C

AAAAAABBBBBCD

##<6+5+1+1>*5*4*3## choice A,B,E

AAAAAABBBBCCD

##<6+4+2+1>*5*4*3*2## choice A,B,C,D

AAAAAABBBCCCD

##<6+3+3+1>*5*4*3## choice A,D,E

AAAAAABBBCCDD

##<6+3+2+2>*5*4*3## choice A,B,E

##7-of-a-kind##

In addition to (*)

AAAAAAABBBBBC

##<7+5+1>*5*4*3## choice A,B,C

AAAAAAABBBBCC

##<7+4+2>*5*4*3## choice A,B,C

AAAAAAABBBCCC

##<7+3+3>*5*\ _4C_2## choice A,(BC)

AAAAAAABBBBCC

##<7+4+1+1>*5*4*3## choice A,(BC)

AAAAAAABBBCCD

##<7+3+2+1>*5*4*3*2## choice A,B,C,D

AAAAAAABBCCDD

##<7+2+2+2>*5*4## choice A,E

AAAAAAABBBCDE

##<7+3+1+1+1>*5*4## choice A,B

AAAAAAABBCCDE

##<7+2+2+1+1>*5*\ _4C_2## choice A,(BC)

****************************

N(14)

##Two\ 7-of-a-kind##

AAAAAAABBBBBBB

##<7+7>*5*4##... choice A,B

##One\ 7-of-a-kind##

AAAAAAABBBBBBC

##<7+6+1>*5*4*3## choice A,B,C

AAAAAAABBBBBCC

##<7+5+2>*5*4*3## choice A,B,C

AAAAAAABBBBCCC

##<7+4+3>*5*4*3## choice A,B,C

AAAAAAABBBBBCD

##<7+5+1+1>*5*4*3## choice A,B,E

AAAAAAABBBBCCD

##<7+4+2+1>*5*4*3*2## choice A,B,C,D

AAAAAAABBBCCCD

##<7+3+3+1>*5*4*3## choice A,D,E

AAAAAAABBBCCDD

##<7+3+2+2>*5*4*3## choice A,B,E

AAAAAAABBBBCDE

##<7+4+1+1+1>*5*4*3## choice A,B,E

AAAAAAABBBCCDE

##<7+3+2+1+1>*5*4*3## choice A,B,C

AAAAAAABBCCDDE

##<7+2+2+2+1>*5*4## choice A,E

**********************************

Last edited: