First slide

Combinations

Question

The number of positive integral solutions of the inequality 3x + y + z ≤ 30, is

Moderate

A

1115
B

1215
C

1315
D

None of these

Solution

Let w be a non-negative integer such that
3x + y + z + w = 30
Let a = x – 1, b = y – 1, c = z – 1, d = w, then
3a + b + c + d = 25, where a, b, c, d ≥ 0 (1)
Clearly, 0 ≤ a ≤ 8. If a = k, then
b + c + d = 25 – 3k (2)
Number of non-negative integral solutions of equation (2)
$\begin{array}{l} =^{n + r - 1} C_{r} =^{3 + 25 - 3 k - 1} C_{25 - 3 k} \\ =^{27 - 3 k} C_{25 - 3 k} =^{27 - 3 k} C_{2} = \frac{(27 - 3 k) (26 - 3 k)}{2} \\ = \frac{3}{2} (3 k^{2} - 53 k - 234) \end{array}$
$∴ Required number = \frac{3}{2} \sum_{k = 0}^{8} (3 k^{2} - 53 k + 234)$
$= \frac{3}{2} [3 \cdot \frac{8 \times 9 \times 17}{6} - 53 \frac{8 \times 9}{2} + 234 \times 9] = 1215$ .

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