Question

Find the value of $^{10}{C_{10}}$

Answer 1

Hint: Problem can be solved using basic combination formula.
$^n{C_n} = \dfrac{{n!}}{{r!(n - r)!}}$

Complete step by step answer:
$n! = $ Product of n natural numbers
$ = 1 \times 2 \times 3 \times ......... \times n$
Combination : It is number of selection of r objects out of n given objects and is given by
$^n{C_r} = \dfrac{{n!}}{{r!(n - r)!}}$
Let say ABC, are three objects out of which 2 objects need to be selected.
Then, number of selection are $ \to AB,BC,AC = 3$
By formula $^n{C_r} = \dfrac{{n!}}{{r!(n - r)!}} = \dfrac{{3!}}{{2!(3 - 2)!}}$
$ = \dfrac{{3 \times 2 \times 1}}{{2 \times 1 \times 1}} = 3$
So for $^{10}{C_{10}}$
\[^n{C_n} = \dfrac{{n!}}{{n!(n - n)!}} = \dfrac{{n!}}{{n! \times 0!}} = 1\]
Value of $0! = 1$

So $^{10}{C_{10}} = \dfrac{{10!}}{{10!(10 - 10)!}} = 1$.

Note: If A, B, C are three numbers & 2 need to be selected out of A, B, C then AC, CA are considered as one case.