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

Answer

165

Explanation

Let a=2p+1, b=2p+1,c=2r+1,d=2s+1 where p,q,r,s are non-negative integers. Therefore,
2p+1+2p+1+2r+1+2s+1=20
or p+q+r+s=8
Therefore, the required number is equal to the number of non-negative intergral solutions of p+q+r+s=8, which is given by `.(8+4-1)C_+(4-1)= .^(11)C_(3)=165`.