Give a generating function where the coefficient of x81 is t

Give a generating function where the coefficient of x^81 is the number of ways you can give 81 identical clocks to 5 collectors (collectors are named A, B, C, D, and E). Collector A will only accept clocks in pairs. Collector B only likes prime numbers and will not accept a composite number of clocks. Collectors C and D will only take multiples of 5 and collector E insists on taking at least two clocks.

Solution

x1 + x2 + x3 + x4 + x5 = 81

x1 is a 2 multiple

x2 is a prime number

x3 and x4 are multiples of 5

x5 >= 2

Now give 2 clocks to x5 we have sum = x1 + x2 + x3 + x4 + x5 = 79

every thing is greater than equal to zero and all above conditions apply

So generating function is of the form (1 +x+ x^2 + x^5+ x^3 + x^7 + x^11+ ... powers of all primes)^79

power of x will be coefficient of x5

power of x^2 is coefficient x1

Power of x^5 is the total 5 multiple

And the rest will be power of primes.