Question

Evaluate the expression $$^5{C_2}$$?

Answer 1

Hint: Here the given question is based on the concept of combination. We have evaluated the given expression which is in the form of a combination. So for this, we will use the formula of combination i.e., $$^n{C_r} = \dfrac{{n!}}{{\left( {n - r} \right)!r!}}$$ where, ‘$$n$$’ is the total number of objects and ‘$$r$$’ is the number of object taken at each time. On substituting and simplifying the formula by using a factorial expansion then we get the required solution.

Complete step-by-step answer:
Combination is defined as “the arrangement of ways to represent a group or number of objects by selecting them in a set and forming the subsets”
The formula used to calculate the combination is: $$^n{C_r} = \dfrac{{n!}}{{\left( {n - r} \right)!r!}}$$----(1)
Where, ‘$$n$$’ is the number of items you have to choose from (total number of objects) and ‘$$r$$’ is the number of items you're going to select.
Now consider the question,
We need to evaluate the expression $$^5{C_2}$$
Compare the given expression with standard combination expression i.e., $$^n{C_r}$$
Here, $$n = 5$$ and $$r = 2$$
By the formula of combination, we can written as
$$ \Rightarrow \,\,{\,^5}{C_2} = \dfrac{{5!}}{{\left( {5 - 2} \right)!2!}}$$
$$ \Rightarrow \,\,\,\,\dfrac{{5!}}{{3!\, \times 2!}}$$
Expand the above expression by using a factorial expansion, then we have
$$ \Rightarrow \,\,\,\,\dfrac{{5 \times 4 \times 3 \times 2 \times 1}}{{3 \times 2 \times 1\, \times 2 \times 1\,}}$$
On cancelling the like terms in both numerator and denominator, then we get
$$ \Rightarrow \,\,\,\,5 \times 2$$
$$ \Rightarrow \,\,\,\,10$$
Hence, the value of $$^5{C_2} = 10$$.
So, the correct answer is “10”.

Note: In combinations each of the different selections made by taking some or all of a number of objects irrespective of their arrangement. Remember, factorial is the continued product of first n natural numbers is called the “n factorial “ and it represented by$$n! = \left( {n - 1} \right) \cdot \left( {n - 2} \right) \cdot \left( {n - 3} \right).....3 \cdot 2 \cdot 1$$.