Properties of ⁿCr – II
3) ⁿCr + ⁿC(r-1) = (ⁿ⁺1)Cr
L.H.S = ⁿ C r + ⁿ C (r -1)
\(^{n}{{C}_{r}}=\frac{n!}{(n-r)!r!}\).
\(^{n}{{C}_{r-1}}=\frac{n!}{(n-(r-1))!(r-1)!}\).
\(=\frac{n!}{(n-r)!r!}+\frac{n!}{(n-(r-1))!(r-1)!}\).
\(=\frac{n!}{(n-r)!r!}+\frac{n!}{(n-r+1)!(r-1)!}\).
\(=\frac{n!}{(r-1)!(n-r)!}\left[ \frac{1}{r}+\frac{1}{n-r+1} \right]\).
\(=\frac{n!}{(r-1)!(n-r)!}\left[ \frac{n-r+1+r}{r(n-r+1)} \right]\).
\(=\frac{n!}{(r-1)!(n-r)!}\left[ \frac{n+1}{r(n-r+1)} \right]\).
\(=\frac{(n+1)n!}{r(r-1)!(n-r+1)(n-r)!}\).
\(=\frac{(n+1)!}{r!(n-r+1)!}{{=}^{n+1}}{{C}_{r}}\).
Hence L.H.S = R. H. S
ⁿCr + ⁿC(r-1) = ⁿ⁺1Cr
4. rⁿCr = n(ⁿ ⁻ ¹) C(r-1)
L. H. S = rⁿCr
\(r\ {{C}_{r}}\ =\ r\ \times \frac{n!}{(n-r)!r!}\).
\(=\ r\ \times \frac{n!}{(n-r)!r(r-1)!}\).
\(=\ \frac{n!}{(n-r)!(r-1)!}\).
\(=\ \frac{n(n-1)!}{(n-r)!(r-1)!}\).
\(=\ n\times \frac{(n-1)!}{(n-r)!(r-1)!}\).
= n(ⁿ ⁻ ¹) C(r-1)
Hence L.H.S = R. H. S.