Find the number of positive integral solutions for the equation x + y + z = 20 where x > 2, y > 3 and z > 4.
Explanation:
Minimum value of x = 3, y = 4 and z = 5.
Let x = a + 3, y = b + 4 and z = c + 5 [a, b, c ≥ 0]
⇒ (a + 3) + (b + 4) + (c + 5) = 20
⇒ a + b + c = 8
∴ Number of integral solutions for x + y + z = 20 is same as number of integral solutions for a + b + c = 8.
⇒ Number of integral solutions for a + b + c = 8 = 8+3-1C3-1 = 10C2 = 45
Hence, 45.
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