Let n N with n 3. Then n 3 can be expressed as a polynomial

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Solution

Q.
Let $n \in N$ with $n \geq 3$ . Then $(\binom{n}{3})$ can be expressed as a polynomial

1 $\frac{1}{6} n^{3} - \frac{1}{2} n^{2} + \frac{1}{3} n$

2 $n^{3} - n^{2} + n$

3 $n^{3} - 3 n^{2}$

4 $n^{3} - 3 n^{2} + n$

Ans.
(1)

$(\binom{n}{3}) = {}^{n}C_{3} = \frac{n (n - 1) (n - 2)}{3!} = \frac{n (n - 1) (n - 2)}{6}$

$= \frac{n^{3} - 3 n^{2} + 2 n}{6} = \frac{1}{6} n^{3} - \frac{1}{2} n^{2} + \frac{n}{3}$

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