Number of integral solutions to the equation x + y + z = 21 , where x ≥ 1 , y ≥ 3 , z ≥ 4 , is equal to _____ .

Question

Number of integral solutions to the equation x + y + z = 21 , where x ≥ 1 , y ≥ 3 , z ≥ 4 , is equal to _____ .,

MARKS App · Accepted Answer

We have, x ≥ 1 , y ≥ 3 , z ≥ 4 ⇒ x - 1 ≥ 0 , y - 3 ≥ 0 , z - 4 ≥ 0 So, x + y + z = 21 ⇒ x - 1 + y - 3 + z - 4 = 13 So, number of non-negative integral solutions is = C 3 - 1 13 + 3 - 1 = C 2 15 = 15 × 14 2 = 105

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