IIT-JEE 2010 · Paper 2 · Mathematics Q20

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MathematicsSingle correct+5 / -2Question 20 of 57

For $r=0,1,\ldots,10$ , let $A_r$ , $B_r$ and $C_r$ denote, respectively, the coefficient of $x^r$ in the expansions of $(1+x)^{10}$ , $(1+x)^{20}$ and $(1+x)^{30}$ . Then $\sum_{r=1}^{10}A_r\,(B_{10}B_r-C_{10}A_r)$ is equal to

B_{10}-C_{10}

A_{10}\left(B_{10}^2-C_{10}A_{10}\right)

C_{10}-B_{10}

Answer and worked solution

Official answer: D

Vandermonde's identity. $A_r=\binom{10}r$ , $B_r=\binom{20}r$ , $C_r=\binom{30}r$ .
$\sum_{r=0}^{10}\binom{10}r\binom{20}r=\sum_{r=0}^{10}\binom{10}r\binom{20}{20-r}=\binom{30}{20}=C_{10},\qquad\sum_{r=0}^{10}\binom{10}r^2=\binom{20}{10}=B_{10}.$
Dropping the $r=0$ terms (each equal to 1): $\sum_{r=1}^{10}A_rB_r=C_{10}-1$ and $\sum_{r=1}^{10}A_r^2=B_{10}-1$ .

The sum.
$B_{10}(C_{10}-1)-C_{10}(B_{10}-1)=C_{10}-B_{10}.$

(D)

Topics:Properties of binomial coefficients

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IIT-JEE 2010 · Paper 2, Mathematics question 20