Number of integral solution to the equation x + y + z = 21, where x 1 , y 3 , z 4 , is equal to __________ . [2023]

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Solution

Q.
Number of integral solution to the equation x + y + z = 21, where $x \geq 1, y \geq 3, z \geq 4$ , is equal to __________ . [2023]

Ans.
(105)

$Given, x + y + z = 21, x \geq 1, y \geq 3, z \geq 4$

$We have, x \geq 1, y \geq 3, z \geq 4$

$Let x - 1 = X,$ i.e., $x = 1 + X; y - 3 = Y,$ i.e., $y = Y + 3; z - 4 = Z,$ i.e., $z = Z + 4$ $∵$ $x + y + z = 21$

$\Rightarrow 1 + X + Y + 3 + Z + 4 = 21$

$∴$ $X + Y + Z = 21 - 8 = 13$

$∴$ $Number of non-negative integral solutions$

$= {}^{13 + 3 - 1}{C_{3 - 1}} = {}^{15}{C_{2}} = \frac{15 \times 14}{2} = 105$

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