If displaystyle∑r=2n ( n-2 Cr-2/ n Cr)=20, then n equals

Question

If displaystyle∑r=2n ( n-2 Cr-2/ n Cr)=20, then n equals (A) 39 (B) 49 (C) 59 (D) 69

Tardigrade Admin · Accepted Answer

displaystyle∑ r =2 n ( n -2 C r -2/( n ( n -1)) r ( r -1) ⋅ n -2 C r -2)=20 displaystyle∑ r =2 n ( r ( r -1)/ n ( n -1))=20 (1/ n ( n -1)) displaystyle∑ r =2 n ( r 2- r )=20 (1/ n ( n -1))(( n ( n +1)(2 n +1)/6)-1-(( n ( n +1)/2)-1))=20 ⇒ n =59

Q. If r=2∑n​nCr​n−2Cr−2​​=20, then n equals

39

49

59

69

r=2∑n​r(r−1)n(n−1)​⋅n−2Cr−2​n−2Cr−2​​=20 r=2∑n​n(n−1)r(r−1)​=20 n(n−1)1​r=2∑n​(r2−r)=20 n(n−1)1​(6n(n+1)(2n+1)​−1−(2n(n+1)​−1))=20 ⇒n=59

Q. If $r = 2 \sum n \frac{^{n - 2} C _{r - 2}}{^{n} C _{r}} = 20$ , then $n$ equals