The number of elements in the set S = { ( x , y , z ) : x , y , z ? Z , ? x + 2 y + 3 z = 42 , ? x , y , z ? 0 } equals _______ .JEE Main Mathematics Sets Relations And Functions

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Solution

Q.
The number of elements in the set $S = {(x, y, z) : x, y, z \in Z, x + 2 y + 3 z = 42, x, y, z \geq 0}$ equals _______ . [2024]

Ans.
(169)

$x + 2 y + 3 z = 42$

At $z = 0, x + 2 y = 42$

$y$ can take all value from 0 to 21

Hence 22 solutions.

Similarly,

at $z = 1, x + 2 y = 39$    $\to$    $20 solutions$

at $z = 2, x + 2 y = 36$    $\to$    $19 solutions$

at $z = 3, x + 2 y = 33$    $\to$    $17 solutions$

at $z = 4, x + 2 y = 30$    $\to$    $16 solutions$

at $z = 5, x + 2 y = 27$    $\to$    $14 solutions$

at $z = 6, x + 2 y = 24$    $\to$    $13 solutions$

at $z = 7, x + 2 y = 21$    $\to$    $11 solutions$

at $z = 8, x + 2 y = 18$    $\to$    $10 solutions$

at $z = 9, x + 2 y = 15$    $\to$    $8 solutions$

at $z = 10, x + 2 y = 12$ $\to$ $7 solutions$

at $z = 11, x + 2 y = 9$    $\to$    $5 solutions$

at $z = 12, x + 2 y = 6$    $\to$    $4 solutions$

at $z = 13, x + 2 y = 3$    $\to$    $2 solutions$

at $z = 14, x + 2 y = 0$      $\to$    $1 solutions$

Total cases = 169 solutions.

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