Paragraph: Given that C(n, r): C(n, r+1)=1: 2 and C(n, r+1): C(n, r+2)=2: 3 . Question: What is P (n, r): C (n, r) equal to?

Step-by-step solution

aligned & ^n C _r ^n C _ r+1 = 1 2 (given) & n! r!(n-r)! n! (r+1)!(n-r-1)! = 1 2 & (r+1)!(n-r-1)! r!(n-r)! = 1 2 & (r+1) r!(n-r-1)! r!(n-r)(n-r-1)! = 1 2 & n-r=2(r+1) & n-r=2 r+2 & n=3 r+2 ..(i) & Again, & ^n C_ r+1 ^n C_ r+2 = 2 3 aligned aligned & n! (r+1)!(n-r-1)! n! (r+2)!(n-r-2)! = 2 3 & (r+2)!(n-r-2)! (r+1)!(n-r-1)! = 2 3 & (r+2)(r+1)!(n-r-2)! (r+1)!(n-r-1)(n-r-2)! = 2 3 & 2(n-r-1)=3(r+2) & 2 n-2 r-2=3 r+6 & 2 n=5 r+8...(ii) aligned Multiply equation (i) by 2 2 n=6 r+4 ...(iii) From equations (ii) and (iii) a