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

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Q. If m,n,r are integers such that $r < m,n,$ then $\,{}^mC_r+\,{}^mC_{r-1}\,{}^nC_1+\,{}^mC_{r-2}\,{}^nC_2+.....+\,{}^mC_1\,{}^nC_{r-1}+\,{}^nC_r$

Binomial Theorem

Solution:

We know that
$\left(1 + x\right)^{m}\, \left(1 + x\right)^{n}$
$= \left(^{m}C_{0}+\,{}^{m}C_{1}x+\,{}^{m}C_{2}x^{2}+..... + \,{}^{m}C_{m}x^{m}\right)$.
$\left(^{n}C_{0}+\,{}^{n}C_{1}x+\,{}^{n}C_{2}x^{2}+..... +\,{}^{n}C_{n}x^{n}\right)$
Equating co-eff. of $x^{r}$ on both sides, we get
$^{m}C_{r} +\,{}^{m}C_{r-1}\,{}^{n}C_{1}+\,{}^{m}C_{r-2}\,{}^{n}C_{2}+..... +\,{}^{n}C_{r}$
$=\,{}^{m+n}C_{r} \quad\left[\because L.H.S. = \left(1+x\right)^{m+n}\right]$