By Pascal's identity, (\binom{5}{2}+\binom{5}{3}) is equal to which expression?

Correct answer: (\binom{6}{3})

Direct answer: Option A, \(\binom{6}{3}\). Pascal's identity says \(\binom{n}{r}+\binom{n}{r+1}=\binom{n+1}{r+1}\). Compare the expression with this pattern: take n=5 and r=2. Then \(\binom{5}{2}+\binom{5}{3}=\binom{6}{3}\). Numerically, the left side is 10+10=20, and \(\binom{6}{3}=20\), which confirms the identity. Option A is correct. Option B, \(\binom{6}{2}\), equals 15, so it does not match. Option C, \(\binom{5}{5}\), equals 1 and is unrelated to the required adjacent pair. Option D, \(\binom{10}{5}\), equals 252 and comes from a different combination problem. The important feature is that the lower numbers 2 and 3 are consecutive, and the upper number increases from 5 to 6. Memory cue: adjacent combinations in one row combine into the next row of Pascal's triangle.