First slide

Combinations

Question

If $n =^{m} C_{2}$ the value of $^{n} C_{2}$ is given by

Moderate

A

$^{m + 1} C_{4}$
B

$m - 1_{C_{4}}$
C

$^{m + 2} C_{4}$
D

$3^{m + 3} C_{4}$

Solution

We have,
$\begin{aligned} n =^{m} C_{2} = \frac{m (m - 1)}{2} \\ ∴^{n} C_{2} = \frac{n (n - 1)}{2} \end{aligned}$
$\begin{array}{l} \Rightarrow^{n} C_{2} = \frac{m (m - 1)}{4} \{\frac{m (m - 1)}{2} - 1\} \\ \Rightarrow^{n} C_{2} = \frac{1}{8} m (m - 1) (m^{2} - m - 2) \\ \Rightarrow^{n} C_{2} = \frac{1}{8} m (m - 1) (m - 2) (m + 1) \\ \Rightarrow^{n} C_{2} = 3 \{\frac{1}{24} (n + 1) m (m - 1) (m - 2)\} = 3^{m + 3} C_{4} \end{array}$

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