First slide

Combinations

Question

The number of ways in which 16 identical things can be distributed among 4 persons if each person gets at least 3 things, is

Moderate

A

33
B

35
C

38
D

None of these

Solution

Required number = coefficient of x¹⁶ in (x³ + x⁴ + … + x⁷)⁴
= coefficient of x¹⁶ in x¹² (1 + x + … + x⁴)
$\begin{matrix} = coefficient of x^{4} in {(\frac{1 - x^{5}}{1 - x})}^{4} \\ = coefficient of x^{4} in [{(1 - x^{5})}^{4} (1 - x)^{- 4}] \\ = coefficient of x^{4} in [(1 - 4 x^{5} + 6 x^{10} - \dots) (1 - x)^{- 4}] \\ = coefficient of x^{4} in (1 - x)^{- 4} \\ =^{7} C_{4} = \frac{4 \cdot 5 \cdot 6 \cdot 7}{4!} = 35 \end{matrix}$

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