Question

# If x, y, z are integers and x 0, y 1, z 2, x + y + z = 15, then the number of values of the ordered triplet (x, y, z) is -

- 91
- 455
^{17}C_{15}
- None of these

^{17}C_{15}## The correct answer is: 91

### Let y = p + 1 and z = q + 2.

Then x 0, p 0, q 0 and x + y + z = 15

x + p + q = 12

The reqd. number of values of (x, y, z) and hence of (x, p, q)

= No. of non-negative integral solutions of x + p + q= 12

= Coeff. of x^{12} in (x^{0} + x^{1} + x^{2} + ……)^{3}

= Coeff. of x^{12} in (1 – x)^{–3}

= Coeff. of x^{12} in [^{2}C_{0} + ^{3}C_{1} x + ^{4}C_{2} x^{2} + ….]

= ^{14}C_{12} = = = 91.

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