Question

What is the value of $^n{C_n}$?
A. Zero
B. 1
C. n
D. n!

Answer 1

Hint: In the above question, we need to calculate the value of $^n{C_n}$. We know that the formula of $^n{C_r}$= $\dfrac{{n!}}{{(n - r)!r!}}$. We will put the $^n{C_n}$in this formula where r = n. Let’s see how we can calculate its value.

Complete step by step solution:
We know that $^n{C_r}$= $\dfrac{{n!}}{{(n - r)!r!}}$
Now we will put $^n{C_n}$ in the above formula to calculate its value.
$^n{C_n}$= $\dfrac{{n!}}{{(n - n)!n!}} = \dfrac{{n!}}{{0!n!}} = 1$
Hence, the value of $^n{C_n}$ is 1.

So, the correct answer is “Option B”.

Note: A combination is the way of selecting the objects or collection, in such a way that the order of the objects does not matter. Formula of combination is $^n{C_r}$= $\dfrac{{n!}}{{(n - r)!r!}}$. Let’s see an example of a combination.
Question- How many combinations can we write using the vowels of the word GREAT.
Solution- Number of vowels in the word GREAT, r= 2
Total number of words, n = 5
Number of combinations$^5{C_2}$ = $\dfrac{{5!}}{{(5 - 2)!2!}} = \dfrac{{5!}}{{3!2!}} = \dfrac{{5 \times 4 \times 3!}}{{3! \times 2 \times 1}} = 10$(Ans.)
Hence, the number of combinations is 10.