Binomial Coefficient C(n, r)

It counts the number of ways to choose $r$ objects from $n$ without order. The formula is $\binom{n}{r} = \frac{n!}{r!(n-r)!}$. So $\binom{5}{2} = 10$ and $\binom{6}{3} = 20$. These coefficients appear in the binomial expansion $(a+b)^n$.

Slide $n$ and $r$. The lab returns $\binom{n}{r}$ and shows it inside Pascal's triangle. Try $n = 4$, $r = 2$ to see $6$, the middle of row $4$. Notice $\binom{n}{r} = \binom{n}{n-r}$ — the symmetry of the row.

The binomial theorem $(a+b)^n = \sum_{r=0}^{n}\binom{n}{r}a^{n-r}b^r$ is a JEE staple. Coefficient extraction, middle term in $(1+x)^{10}$, and probability of $r$ heads in $n$ coin flips all use $\binom{n}{r}$. Mastering the lab speeds up dozens of problem types.

For small $n \le 8$, Pascal's triangle (each entry sums the two above) is fastest by hand. For $n > 10$, the formula $\frac{n!}{r!(n-r)!}$ wins, though factorials grow fast. The lab computes both ways to help you pick the right tool in an exam.