If x, y, z and t are odd natural numbers such that x + y + z + t = 20, then find the number of values of ordered quadruplet(x, y, z and t).?

Transform the problem by writing

x = 2 x' + 1

y = 2 y' + 1

z = 2 z' + 1

t = 2 t' + 1

=> x + y + z + t = 2 (x' + y' + z' + t') + 4 = 20

=> x' + y' + z' + t' = 8

This a stars and bars problem.

We have 8 stars to divide under 4 groups.

So we put 3 bars for the subdivision.

=> C(8 + 3, 3) = C(11,3) = 11x10x9/(1x2x3) = 11x5x3 = 165 possibilities.

The answer is 165.

(Note that my y and z are indeterminates and not your x,y,z,t)

The generating function for odd values of each of variable is z^1 + z^3 + z^5 + z^7 + ... = z/(1-z^2) so the ogf for the system is:

z^4/(1-z^2)^4

since there is a known expansion:

y^k/(1-y)^(k+1) = sum(C(n,k)y^n)

replace y with z^2 and k with 3 we have:

z^6/(1-z^2)^4 = sum(C(n,3)(z^2)^n)

z^2(z^4/(1-z^2)^4) = z^2*(sum(C(n,3)z^(2(n-1))))

z^4/(1-z^2)^4 = sum(C(n,3)z^(2(n-1)))

z^4/(1-z^2)^4 = sum(C(m/2 +1,3)z^m)

the appropriate coefficient in this summation is that of the z^20 term, so m=20:

C(20/2 + 1,3) = C(11,3) = 11*10*9/(3*2) = 165 <----