
How do you use binomial theorem to calculate ${}^{{}_6}{C_{^4}}$ ?
Answer
458.7k+ views
Hint: The binomial theorem tells us to expand the expression of the form ${(a + b)^n}$ . Here we asked to calculate ${}^{{}_6}{C_{^4}}$ by using a binomial theorem for which we use the combination formula. The combination formula is ${}^n{C_{^r}} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}$, 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 ${}^{{}_6}{C_{^4}}$
Now use combination formula to solve
${}^n{C_{^r}} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}$
By comparing ${}^{{}_6}{C_{^4}}$ and ${}^n{C_{^r}}$, we get n = 6 & r = 4
$ \Rightarrow {}^6{C_4} = \dfrac{{6!}}{{4!\left( {6 - 4} \right)!}}$
$ \Rightarrow {}^6{C_4} = \dfrac{{6!}}{{4!2!}}$
By expanding factorial, we get
\[ \Rightarrow {}^6{C_4} = \dfrac{{1 \times 2 \times 3 \times 4 \times 5 \times 6}}{{1 \times 2 \times 3 \times 4 \times 1 \times 2}}\]
In the above expansion \[1 \times 2 \times 3 \times 4\]is common in numerator and denominator so we cancel it, then we get
\[ \Rightarrow \dfrac{{5 \times 6}}{{1 \times 2}}\]
With basic mathematical calculation, we get
\[ \Rightarrow \dfrac{{30}}{2}\]
Let us divide the term and we get
\[ \Rightarrow 15\]
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.
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 ${}^{{}_6}{C_{^4}}$
Now use combination formula to solve
${}^n{C_{^r}} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}$
By comparing ${}^{{}_6}{C_{^4}}$ and ${}^n{C_{^r}}$, we get n = 6 & r = 4
$ \Rightarrow {}^6{C_4} = \dfrac{{6!}}{{4!\left( {6 - 4} \right)!}}$
$ \Rightarrow {}^6{C_4} = \dfrac{{6!}}{{4!2!}}$
By expanding factorial, we get
\[ \Rightarrow {}^6{C_4} = \dfrac{{1 \times 2 \times 3 \times 4 \times 5 \times 6}}{{1 \times 2 \times 3 \times 4 \times 1 \times 2}}\]
In the above expansion \[1 \times 2 \times 3 \times 4\]is common in numerator and denominator so we cancel it, then we get
\[ \Rightarrow \dfrac{{5 \times 6}}{{1 \times 2}}\]
With basic mathematical calculation, we get
\[ \Rightarrow \dfrac{{30}}{2}\]
Let us divide the term and we get
\[ \Rightarrow 15\]
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.
Recently Updated Pages
The correct geometry and hybridization for XeF4 are class 11 chemistry CBSE

Water softening by Clarks process uses ACalcium bicarbonate class 11 chemistry CBSE

With reference to graphite and diamond which of the class 11 chemistry CBSE

A certain household has consumed 250 units of energy class 11 physics CBSE

The lightest metal known is A beryllium B lithium C class 11 chemistry CBSE

What is the formula mass of the iodine molecule class 11 chemistry CBSE

Trending doubts
Is Cellular respiration an Oxidation or Reduction class 11 chemistry CBSE

In electron dot structure the valence shell electrons class 11 chemistry CBSE

What is the Pitti Island famous for ABird Sanctuary class 11 social science CBSE

State the laws of reflection of light

One Metric ton is equal to kg A 10000 B 1000 C 100 class 11 physics CBSE

Difference Between Prokaryotic Cells and Eukaryotic Cells
