If m,n,r are integers such that r

Question

If m,n,r are integers such that r < m,n, then mCr+ mCr-1 nC1+ mCr-2 nC2+.....+ mC1 nCr-1+ nCr (A) ( nCr)2 (B)  m+nCr (C)  m+nCr+ mCr+ nCr (D) none of these.

Tardigrade Admin · Accepted Answer

We know that (1 + x)m (1 + x)n = (mC0+ mC1x+ mC2x2+..... + mCmxm). (nC0+ nC1x+ nC2x2+..... + nCnxn) Equating co-eff. of xr on both sides, we get mCr + mCr-1 nC1+ mCr-2 nC2+..... + nCr = m+nCr [∵ L.H.S. = (1+x)m+n]

Q. If m,n,r are integers such that $r < m, n,$ then $^{m} C_{r} +^{m} C_{r - 1}^{n} C_{1} +^{m} C_{r - 2}^{n} C_{2} + ..... +^{m} C_{1}^{n} C_{r - 1} +^{n} C_{r}$

$(^{n} C_{r})^{2}$

$^{m + n} C_{r}$

$^{m + n} C_{r} +^{m} C_{r} +^{n} C_{r}$

none of these.

Q. If m,n,r are integers such that r<m,n, then mCr​+mCr−1​nC1​+mCr−2​nC2​+.....+mC1​nCr−1​+nCr​

(nCr​)2

m+nCr​

m+nCr​+mCr​+nCr​

none of these.

We know that (1+x)m(1+x)n =(mC0​+mC1​x+mC2​x2+.....+mCm​xm). (nC0​+nC1​x+nC2​x2+.....+nCn​xn) Equating co-eff. of xr on both sides, we get mCr​+mCr−1​nC1​+mCr−2​nC2​+.....+nCr​ =m+nCr​[∵L.H.S.=(1+x)m+n]

Q. If m,n,r are integers such that $r < m, n,$ then $^{m} C_{r} +^{m} C_{r - 1}^{n} C_{1} +^{m} C_{r - 2}^{n} C_{2} + ..... +^{m} C_{1}^{n} C_{r - 1} +^{n} C_{r}$

$(^{n} C_{r})^{2}$

$^{m + n} C_{r}$

$^{m + n} C_{r} +^{m} C_{r} +^{n} C_{r}$