If ^n C_ r-1 =330, ^n C_r=462 , and ^n C_ r+1 =462 , then r is equal to

Step-by-step solution

Given, ^n C_ r-1 =330, ^n C_r=462 and ^n C_ r+1 =462 Now, ^n C_ r+1 ^n C_r =1 aligned & n ! (r+1) !(n-r-1) ! n ! r !(n-r) ! =1 & r !(n-r)(n-r-1) ! (r+1) r !(n-r-1) ! =1 & n-r r+1 =1 n-2 r=1 aligned Again, ^n C_r ^n C_ r-1 = 462 330 = 77 55 n ! r !(n-r) ! n ! (r-1) !(n-r+1) ! = 77 55 array ll & (r-1) !(n-r+1)(n-r) ! r(r-1) !(n-r) ! = 77 55 & 55 n-132 r+55=0 array From Eqs. (i) and (ii), we get aligned & & 22 r & =110 & & r & =5 aligned