Use ordinary or exponential generating functions to solve th

Use ordinary or exponential generating functions to solve the following problems:

Find the number of 5-combinations of the multiset {4 middot a, 4 middot b, 4 middot c}. List all of them to check your answer.

Solution

total number of given letters =12 {a,a,a,a,b,b,b,b,c,c,c,c}

number of letters in the required combinations =5

The total possible combinations are as follows

1) {a,a,a,a,b}, {a,a,a,a,c},{b,b,b,b,a},{b,b,b,b,c} {c,c,c,c,a}, {c,c,c,c,b}=6

2) {a,a,a,b,b},{a,a,a,c,c} {b,b,b,a,a} ,{b,b,b,c,c} ,{ c,c,c,a,a},{c,c,c,b,b}=6

    Total number of combinations =12