Set theory

Question

I thought it is multi set, each of the 4 places have 6 choices.  so 6⁴ . .?? please clear this out. What is multiset means?? 

Comments

when you are doing 6⁴, you can come up with two sets as shown below,

{1 , 2, 4, 4} and {4, 4, 1 ,2}

though in multisets duplicates are allowed but oreder does not matter in multisets also.

you can find out like this,

CASE1: all four are different (eg: {1,3,4,5}) ->  $\binom{6}{4}$ = 15

CASE2: one pair of same another pair of different (eg:{1,1,2,4}) ->  $\binom{6}{3}$*3 = 60

CASE3: two pairs of same (eg: {1,1,3,3}) ->  $\binom{6}{2}$ = 15

CASE3: three same and one different (eg: {1,1,1,4})->  $\binom{6}{2}$*2 = 30

CASE4: all same (eg: {2,2,2,2}) ->  $\binom{6}{1}$ = 6

adding all up, you will get 126.

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Please explain how it is "CASE2: one pair of same another pair of different (eg:{1,1,2,4}) ->  (⁶C₃)*3 = 60" ????

Why multiply with 3????

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i will show with reference to this set, {1,1,2,4},.....first select any 3 elements(i.e 1,2,4) in $\binom{6}{3}$ then any one among these 3 can be duplicate(i.e 1,1)..

so total $\binom{6}{3}$*3 ways for performing CASE2

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ok thanx a lot  :) (was not getting this simple thing)

Make ur cmment as answer

Answer 1

The genrating function for this :  (1+x+x²+x³+x⁴+x⁵+ . . . .)⁶

Coefficient of x⁴ in:                (1-x)^-6 = 1+⁶C_¹ x + ⁷C₂ x² +. . . 

                                             = ⁹C₄  = 126