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  4. If x, y, z, w ∈ N, such that each of them gives remainder 3 when divided by 5 , then number of solutions of the equation x+y+z+w=62, is equal to

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Q. If $x, y, z, w \in N$, such that each of them gives remainder 3 when divided by 5 , then number of solutions of the equation $x+y+z+w=62$, is equal to

Permutations and Combinations

Solution:

$x = 5a + 3$
$y = 5b + 3$
$z = 5c + 3$
$w = 5d + 3$
$x + y + z + w =5( a + b + c + d )+12=62=\{ a + b + c + d =10\} $
$\text { so } { }^{10+4-{ }^4 C _{4-1}}={ }^{13} C _3$