The coefficients of x^p and x^q in the expansion of (1+x)^p+q are:

Question

The coefficients of $x^p$ and $x^q$ in the expansion of $(1+x)^{p+q}$ are:

Accepted Answer

Here in this expansion $${\left( {1 + x} \right)^{p + q}}$$

The general term = $${T_{r + 1}} = {}^{p + q}{C_r}.{\left( x \right)^r}$$

$$\therefore$$ ${x^p}$ will be present in the term = $${}^{p + q}{C_p}.{\left( x \right)^p}$$

So coefficient of ${x^p}$ = ${}^{p + q}{C_p}$

And ${x^q}$ will be present in the term = $${}^{p + q}{C_q}.{\left( x \right)^q}$$

$$\therefore$$ coefficient of ${x^q}$ = ${}^{p + q}{C_q}$

We know ${}^n{C_r}$ = ${}^n{C_{n - r}}$

$$\therefore$$ ${}^{p + q}{C_q}$ = $${}^{p + q}{C_{\left( {p + q} \right) - q}}$$ = ${}^{p + q}{C_p}$

So coefficients of ${x^p}$ and ${x^q}$ are equal.

Solutions · 1