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Let \( S=x_{1}+x_{2}+x_{3}+x_{4}+x_{5} \) and \( K=x_{1}+x_{2}+x_{3} \) then
(a) Number of positive integral solutions of the equation \( S \) \( =21 \) is 4845
(b) Number of integral solutions of the equation \( S=21 \) such that \( x_{i} \geq 2 \forall_{i}=1,2,3,4,5 \) is 1365 .
(c) Number of integral solutions of the equation \( S=20 \) such that \( x_{i} \geq 0 \cdot \forall_{i}=1,2,3,4,5 \) is 42404
(d) Number of positive integral solutions \( \left(x_{1}, x_{2}, x_{3}\right. \), \( x_{4}, x_{5} \) ) such that \( S=20 \) and \( K=5 \) is 336
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