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Question

Number of ways in which $$25$$ identical things be distributed among five persons if each gets odd number of things is


A
24C4
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B
12C8
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C
14C10
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D
13C3
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Solution

The correct option is C $$^{14}C_{10}$$
$$\text{Concept: Total number of non-negative integral solution of }x_1+x_2+......+x_r=n \,\text{is}\, {}^{n+r-1}C_{r-1}$$
Also, $$n$$ identical things can be distributed in $$r$$ groups in $${}^{n+r-1}C_{r-1} $$ ways


Let person $$P_i$$ gets $$x_i$$ number of things such that
$$x_1+x_2+x_3+x_4+x_5=25$$
Let $$x_i=2\lambda_i+1$$, where $$\lambda_i\geq 0$$. Then
$$2(\lambda_1+\lambda_2+\lambda_3+\lambda_4+\lambda_5)+5=25$$
or $$\lambda_1+\lambda_2+\lambda_3+\lambda_4+\lambda_5=10$$ (i)
Required number of ways $$=$$ The number of non-negative integral solutions of the equation (i),
which is equal to $$^{10+5-1}C_{5-1} =  ^{14}C_4={}^{14}C_{10}$$.



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