Answer:
C₇,₂ should be written as [tex]\rm ^{7}C_{2}[/tex]
Binomial Coefficient
[tex]\rm ^{n}C_{r}=\dfrac{n!}{r!(n-r)!}[/tex]
The binomial coefficient can be used to calculate the number of ways objects can be arranged.
Imagine you have [tex]n[/tex] objects, but [tex]r[/tex] of these are identical to each other, and the other [tex](n-r)[/tex] are also identical to each other (so two different types of objects).
Therefore, [tex]r[/tex] objects of one type, and [tex](n - r)[/tex] objects of another type can be arranged in [tex]\rm ^{n}C_{r}[/tex] different orders.
The exclamation mark "!" placed after a number means factorial and indicates to multiply all whole numbers from the given number down to 1.
Example: 4! = 4 × 3 × 2 × 1
[tex]\begin{aligned}\implies \rm ^{7}C_{2} & =\dfrac{7!}{2!(7-2)!}\\\\& =\dfrac{7!}{2!\:5!}\\\\& =\dfrac{7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{(2 \times 1)(5 \times 4 \times 3 \times 2 \times 1)}\\\\& =\dfrac{7 \times 6}{2 \times 1}\\\\& =\dfrac{42}{2}\\\\& =21\end{aligned}[/tex]
So if there are 7 objects of two different kinds, where 2 are identical to each other and the other 5 are identical to each other, there are 21 different ways to arrange them.
