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Which identity correctly expresses the symmetry property of binomial coefficients?
A$\binom{n}{r}=\binom{n+1}{r}$
B$\binom{n}{r}=\binom{n}{r+1}$
C$\binom{n}{r}=\binom{r}{n}$
D$\binom{n}{r}=\binom{n}{n-r}$
Answer & Solution
Correct answer: D. $\binom{n}{r}=\binom{n}{n-r}$
Using the factorial formula, $\binom{n}{r}=\frac{n!}{r!(n-r)!}$. Interchanging $r$ and $n-r$ does not change the value, so $\binom{n}{r}=\binom{n}{n-r}$. This is the standard symmetry relation for combinations.