Solveeit Logo
Login

Question

Question: Find the value of \(^{10}{C_{10}}\)...

Find the value of 10C10^{10}{C_{10}}

Explanation

Solution

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

Complete step by step answer:
n!=n! = Product of n natural numbers
=1×2×3×.........×n= 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
nCr=n!r!(n−r)!^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 →AB,BC,AC=3 \to AB,BC,AC = 3
By formula nCr=n!r!(n−r)!=3!2!(3−2)!^n{C_r} = \dfrac{{n!}}{{r!(n - r)!}} = \dfrac{{3!}}{{2!(3 - 2)!}}
=3×2×12×1×1=3= \dfrac{{3 \times 2 \times 1}}{{2 \times 1 \times 1}} = 3
So for 10C10^{10}{C_{10}}
nCn=n!n!(n−n)!=n!n!×0!=1^n{C_n} = \dfrac{{n!}}{{n!(n - n)!}} = \dfrac{{n!}}{{n! \times 0!}} = 1
Value of 0!=10! = 1

So 10C10=10!10!(10−10)!=1^{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.