Properties of ⁿC r – III

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).