Question

How do you evaluate $^{10}{C_3} $

Answer 1

Hint: In order to evaluate the above ,consider $n = 10$ and $r = 3$ and use the formula of $C(n,r)\, = {\,^n}{C_r} = \dfrac{{n!}}{{r!(n - r)!}}$ to find the number of combinations.

Formula used:
$C(n,r)\, = {\,^n}{C_r} = \dfrac{{n!}}{{r!(n - r)!}}$
$p(n,r)\, = {\,^n}{P_r} = \dfrac{{n!}}{{(n - r)!}}$

Complete step by step solution:
Given $^{10}{C_3}$ ,this is of the form ${\,^n}{C_r}$ where $n = 10$ and $r = 3$.
To evaluate this we will use formula of $C(n,r)\, = {\,^n}{C_r} = \dfrac{{n!}}{{r!(n - r)!}}$
So, Putting the value of n and r in the above formula
$
C(n,r)\, = {\,^n}{C_r} = \dfrac{{10!}}{{3!(10 - 3)!}} \\
= \dfrac{{10!}}{{3!7!}} \\
$
$10!$ is equivalent to $10 \times 9 \times 8 \times 7!$
$
= \dfrac{{10 \times 9 \times 8 \times 7!}}{{3!7!}} \\
= \dfrac{{10 \times 9 \times 8 \times {7}!}}{{3!{7}!}} \\
= \dfrac{{10 \times 9 \times 8}}{{3 \times 2}} \\
= 10 \times 3 \times 4 \\
= 120 \\
$
Therefore, value of $^{10}{C_3}$ is equal to $120$

Additional Information:
1.Factorial: The continued product of first n natural numbers is called the “n factorial “ and denoted
by $n!$.
2.Permutation: Each of the arrangements which can be made by taking some or all of number of
things are called permutations. If n and r are positive integers such that $1 \leqslant r \leqslant n$, then the number of all permutations of n distinct or different things, taken r at one time is denoted by the symbol
$p(n,r)\,or{\,^n}{P_r}$.
$p(n,r)\, = {\,^n}{P_r} = \dfrac{{n!}}{{(n - r)!}}$
3.Combinations: Each of the different selections made by taking some or all of a number of objects
irrespective of their arrangement is called a combination. The combinations number of n objects, taken r at one time is generally denoted by
$C(n,r)\,or{\,^n}{C_r}$

Thus, $C(n,r)\,or{\,^n}{C_r}$= Number of ways of selecting r objects from n objects.
$C(n,r)\, = {\,^n}{C_r} = \dfrac{{n!}}{{r!(n - r)!}}$

Note: 1. Factorials of proper fractions or negative integers are not defined. Factorial n defined only for whole numbers.
2.Meaning of Zero factorial is senseless to define it as the product of integers from 1 to zero. So, we
define it as $0! = 1$.
3.Don’t forget to cross-check your answer at least once as it may contain calculation errors.