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Question: How do you use binomial theorem to calculate \({}^{{}_6}{C_{^4}}\) ?...

How do you use binomial theorem to calculate 6C4{}^{{}_6}{C_{^4}} ?

Explanation

Solution

The binomial theorem tells us to expand the expression of the form (a+b)n{(a + b)^n} . Here we asked to calculate 6C4{}^{{}_6}{C_{^4}} by using a binomial theorem for which we use the combination formula. The combination formula is nCr=n!r!(n−r)!{}^n{C_{^r}} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}, where n = number of items, r = how many items are taken at a time. This question requires a basic understanding of how to manipulate factorials.
Then factorial is a product of all positive integers less than or equal to a given positive integer and denoted by that integers and an exclamation point. Then we solve this by basic mathematical calculation and complete step by step explanation.

Complete step-by-step solution:
Let us consider the given value 6C4{}^{{}_6}{C_{^4}}
Now use combination formula to solve
nCr=n!r!(n−r)!{}^n{C_{^r}} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}
By comparing 6C4{}^{{}_6}{C_{^4}} and nCr{}^n{C_{^r}}, we get n = 6 & r = 4
⇒6C4=6!4!(6−4)!\Rightarrow {}^6{C_4} = \dfrac{{6!}}{{4!\left( {6 - 4} \right)!}}
⇒6C4=6!4!2!\Rightarrow {}^6{C_4} = \dfrac{{6!}}{{4!2!}}
By expanding factorial, we get
⇒6C4=1×2×3×4×5×61×2×3×4×1×2\Rightarrow {}^6{C_4} = \dfrac{{1 \times 2 \times 3 \times 4 \times 5 \times 6}}{{1 \times 2 \times 3 \times 4 \times 1 \times 2}}
In the above expansion 1×2×3×41 \times 2 \times 3 \times 4is common in numerator and denominator so we cancel it, then we get
⇒5×61×2\Rightarrow \dfrac{{5 \times 6}}{{1 \times 2}}
With basic mathematical calculation, we get
⇒302\Rightarrow \dfrac{{30}}{2}
Let us divide the term and we get
⇒15\Rightarrow 15

Thus we use binomial theorem to calculate ${}^{{}6}{C{^4}}$$$ = 15$$

Note: This problem needs basic attention on binomial theorem, the combination formula and factorial concept. A binomial expression is an expression containing two terms joined by either addition or subtraction sign. Economists used binomial theorem to count probabilities that depend on numerous and very distributed variables to predict the way the economy will behave in the next few years. To be able to come up with realistic predictions and also in design of infrastructure.