Properties of ⁿC r – III
5.\(\frac{^{n}{{C}_{r}}}{r+1}=\frac{^{n+1}{{C}_{r+1}}}{n+1}\).
Proof: L.H.S = \(\frac{^{n}{{C}_{r}}}{r+1}\).
\(=\frac{\frac{n!}{(n-r)!r!}}{(r+1)}\).
\(=\frac{n!}{(r+1)r!(n-r)!}\).
\(=\frac{n!}{(r+1)r!(n-r)!}\times \frac{n+1}{n+1}\).
\(=\frac{(n+1)n!}{(r+1)r!(n-r)!}\times \frac{1}{n+1}\).
\(=\frac{1}{(n+1)}\times \frac{(n+1)!}{(r+!)!(n-r)!}\).
\(=\frac{^{n+1}{{C}_{r+1}}}{n+1}\).
Hence proved \(\frac{^{n}{{C}_{r}}}{r+1}=\frac{^{n+1}{{C}_{r+1}}}{n+1}\).
6. \(\frac{^{n}{{C}_{r}}}{^{n}{{C}_{r-1}}}=\frac{n-r+1}{r}\).
7. Maximum value of ⁿCr
we can observe that in the list of ⁶C₀, ⁶C₁, ⁶C₂, ⁶C₃, ⁶C₄, ⁶C₅ and ⁶C₆the maximum value is ⁶C₃.
Also, in the list of ⁷C₀, ⁷C₁, ⁷C₂, ⁷C₃, ⁷C₄,⁷C₅, ⁷C₆ and ⁷C₇ the maximum value is ⁷C₃, ⁷C₄.
In general, when n is even, maximum value of ⁿCr is ⁿCn/2 and when n is odd, maximum value of ⁿCr is ⁿC(n-1)/2 (or) ⁿC(n+1)/2.
8. The product of k consecutive positive integers is dividable by k!
Let k consecutive integers be m, m + 1, m + 2, … m + k – 1.
m (m + 1) (m + 2) … (m + k- 1)
\(=\frac{(m-1)!m(m+1)….(m+k-1)}{(m-1)!}\).
\(=\frac{(m+k-1)!}{(m-1)!}\).
\(=k!\ \frac{(m+k-1)!}{(m-1)!k!}\).
= (k!) (m+k-1)Ck
∵ (m+k-1)Ck is an integer, it follows that k! divides m(m + 1) … (m + k – 1).