First slide

Combinations

Question

The number of non-negative integral solutions to the system of equations x + y + z + u + t = 20 and x + y + z = 5 is

Moderate

A

336
B

346
C

246
D

None of these

Solution

Given, x + y + z + u + t = 20 (1)
x + y + z = 5 (2)
Given, system of equations is equivalent to
x + y + z = 5 (3)
and, u + t = 15 (4)
Number of non-negative integral solutions of equation (3)
$=^{n + r - 1} C_{r} =^{3 + 5 - 1} C_{5} =^{7} C_{5}$ ,
Number of non-negative integral solutions of equation (4)
$=^{n + r - 1} C_{r} =^{2 + 15 - 1} C_{15} =^{16} C_{15}$
∴ Required number $=^{7} C_{5} \times^{16} C_{15} = 336$ .

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