Question
The number of integral solutions of the equation x + y + z = 24 subjected to conditions that 1 
x 5, 12 y 18, z –1
- 31
- 30
- 29
- None of these
The correct answer is: 30
Let t = z + 1
Equation reduces to x + y + t = 25
1 x 5, 12 y 18, t 
0
Required number of ways
= Coefficient of x25 in [(x + x2 + x3 + x4 + x5) (x12 + x13 +…..+ x18) (1 + x + x2 + …..)]
= Coefficient of x12 in (1 + x + x2 + x3 + x4) (1 + x + x2 + x3 + x4 + x5 + x6) (1 + x + x2 + …..)
= Coefficient of x12 in 
(1 – x)–1
= Coefficient of x12 in (1 – x5) (1 – x6) (1 – x)–3
= Coefficient of x12 in (1 – x5 – x6 + x11) (1 – x)–3
= 12+3–1C3–1 – 7+3–1C3–1 – 6+3–1C3–1 + 1+3–1C3–1
= 14C2 – 9C2 – 8C2 + 3C2
= 91 – 36 – 28 + 3 = 30.
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