## Team Organization

A technical lead is organizing a team of engineers for a new project. With current budget constraints, they can only choose 4 engineers from a pool of 10. How many different ways can the technical lead create a team?

##
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__**Hint**

**Hint**

Unlike a permutation where a particular sequence order is considered, a combination is needed to find the number of groups because order is not important.

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__**Hint 2**

**Hint 2**

$$$C_{(n,r)}=\frac{P_{(n,r)}}{r!}=\frac{n!}{[r!(n-r)!]}$$$

where
$$C_{n,r}$$
is the number of different combinations of
$$n$$
distinct objects taken
$$r$$
at a time, and
$$P$$
is the number of different permutations.

Unlike a permutation where a particular sequence order is considered, a combination is needed to find the number of groups because order does not impact how teams are formed:

$$$C_{(n,r)}=\frac{P_{(n,r)}}{r!}=\frac{n!}{[r!(n-r)!]}$$$

where
$$C_{n,r}$$
is the number of different combinations of
$$n$$
distinct objects taken
$$r$$
at a time, and
$$P$$
is the number of different permutations. Thus, the number of possible team formations based on the problem statement is:

$$$C_{(10,4)}=\frac{10!}{4![(10-4)!]}=\frac{10!}{4!(6!)}=\frac{10\cdot 9\cdot 8\cdot 7\cdot 6!}{4\cdot 3\cdot 2\cdot 1\cdot (6!)}=\frac{5,040}{24}=210$$$

210