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### 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 x^{25} in [(x + x^{2} + x^{3} + x^{4} + x^{5}) (x^{12} + x^{13} +…..+ x^{18}) (1 + x + x^{2} + …..)]

= Coefficient of x^{12} in (1 + x + x^{2} + x^{3} + x^{4}) (1 + x + x^{2} + x^{3} + x^{4} + x^{5} + x^{6}) (1 + x + x^{2} + …..)

= Coefficient of x^{12} in (1 – x)^{–1}

= Coefficient of x^{12} in (1 – x^{5}) (1 – x^{6}) (1 – x)^{–3}

= Coefficient of x^{12} in (1 – x^{5} – x^{6} + x^{11}) (1 – x)^{–3}

= ^{12+3–1}C_{3–1} – ^{7+3–1}C_{3–1} – ^{6+3–1}C_{3–1} + ^{1+3–1}C_{3–1}

= ^{14}C_{2} – ^{9}C_{2} – ^{8}C_{2} + ^{3}C_{2}

= 91 – 36 – 28 + 3 = 30.

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