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Question: How do you evaluate \(^{10}{C_3} \)...

How do you evaluate 10C3^{10}{C_3}

Explanation

Solution

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

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

Complete step by step solution:
Given 10C3^{10}{C_3} ,this is of the form  nCr{\,^n}{C_r} where n=10n = 10 and r=3r = 3.
To evaluate this we will use formula of C(n,r) = nCr=n!r!(n−r)!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) = nCr=10!3!(10−3)! =10!3!7!  C(n,r)\, = {\,^n}{C_r} = \dfrac{{10!}}{{3!(10 - 3)!}} \\\ = \dfrac{{10!}}{{3!7!}} \\\
10!10! is equivalent to 10×9×8×7!10 \times 9 \times 8 \times 7!
=10×9×8×7!3!7! =10×9×8×7!3!7! =10×9×83×2 =10×3×4 =120  = \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 10C3^{10}{C_3} is equal to 120120

Additional Information:
1.Factorial: The continued product of first n natural numbers is called the “n factorial “ and denoted
by n!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⩽r⩽n1 \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 nPrp(n,r)\,or{\,^n}{P_r}.
p(n,r) = nPr=n!(n−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 nCrC(n,r)\,or{\,^n}{C_r}

Thus, C(n,r) or nCrC(n,r)\,or{\,^n}{C_r}= Number of ways of selecting r objects from n objects.
C(n,r) = nCr=n!r!(n−r)!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!=10! = 1.
3.Don’t forget to cross-check your answer at least once as it may contain calculation errors.