JEE Main Mathematics Question (2026) — Solution

Step-by-step solution

The general term of the product is T_r = 1 ^ 15 C_ r-1 + 1 ^ 15 C_r for r = 1, 2, , 13. Using the formula ^ n C_r = n r ^ n-1 C_ r-1 , we have ^ 15 C_r = 15 r ^ 14 C_ r-1 and ^ 15 C_ r-1 = 15 15-(r-1) ^ 14 C_ r-1 = 15 16-r ^ 14 C_ r-1 . Substituting these into T_r: T_r = 16-r 15 ^ 14 C_ r-1 + r 15 ^ 14 C_ r-1 = 16-r+r 15 ^ 14 C_ r-1 = 16 15 ^ 14 C_ r-1 . The product is _ r=1 ^ 13 T_r= _ r=1 ^ 13 16 15 ^ 14 C_ r-1 = 16^ 13 15^ 13 _ r=1 ^ 13 ^ 14 C_ r-1 . The denominator of the product is 15^ 13 (^ 14 C_0 ^ 14 C_1 ^ 14 C_ 12 ). Comparing this with the given expression ^ 13 ^ 14 C_0 ^ 14 C_1 ^ 14 C_ 12 , we get: ^ 13 = 16^ 13 15^ 13 = 16 15 . We need to find the value of 30 : 30 = 30 16 15 = 2 16 = 32.