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JEE Main 2026 · 2 Apr · Shift 2, Mathematics question 6

If for 3≤r≤303\le r\le30, (30C30−r)+3(30C31−r)+3(30C32−r)+(30C33−r)=mCr\left({}^{30}C_{30-r}\right)+3\left({}^{30}C_{31-r}\right)+3\left({}^{30}C_{32-r}\right)+\left({}^{30}C_{33-r}\right)={}^mC_r, then mm equals :

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Answer and worked solution

Official answer: C

Rewrite. 30C30−r=30Cr{}^{30}C_{30-r}={}^{30}C_r, 30C31−r=30Cr−1{}^{30}C_{31-r}={}^{30}C_{r-1}, etc. The sum is
30Cr+3 30Cr−1+3 30Cr−2+30Cr−3.{}^{30}C_r+3\,{}^{30}C_{r-1}+3\,{}^{30}C_{r-2}+{}^{30}C_{r-3}.

Vandermonde. This is the coefficient of xrx^r in (1+x)30(1+x)3(1+x)^{30}(1+x)^3, i.e. 33Cr{}^{33}C_r. So m=33m=33.

(C)

Topics:Properties of binomial coefficients
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