The coefficients of $${x^p}$$ and $${x^q}$$ in the expansion of $${\left( {1 + x} \right)^{p + q}}$$ are
B
equal with opposite signs
C
reciprocals of each other
Explanation
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.
