If 15C8 + 15C7 = nCr, then the values of n and r are: Here nCr denotes the…

Pascal's identity states that for integers n ≥ r ≥ 1, nCr + nCr − 1 = n + 1Cr. In words: when two binomial coefficients share the same upper index and their lower indices differ by exactly 1, their sum is a single binomial coefficient whose upper index is one greater and whose lower index is the larger of the two.

Applying this to the given sum:

1. Both terms on the left share the upper index 15, and their lower indices 8 and 7 differ by exactly 1, so the identity applies with n = 15 and r = 8, because r − 1 = 7.

2. Substituting into the identity: 15C8 + 15C7 = 15C8 + 15C8 − 1 = 15 + 1C8 = 16C8.

3. Comparing 16C8 with nCr gives n = 16 and r = 8.

Check by direct evaluation: 15C8 = 6435 and 15C7 = 6435, so the left-hand side equals 12870; and 16C8 = 12870, which confirms the result.

Contrast with the other pairs:

• 16C7 and 16C9 are both equal to 11440, because nCr = nCn − r holds for every binomial coefficient; 11440 is not 12870.

• 30C15 = 155117520; the identity raises the upper index by exactly one, so adding the two upper indices (15 + 15 = 30) is not what it does.