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Question: What is the value of \(^n{C_n}\)? A. Zero B. 1 C. n D. n!...

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

Explanation

Solution

In the above question, we need to calculate the value of nCn^n{C_n}. We know that the formula of nCr^n{C_r}= n!(n−r)!r!\dfrac{{n!}}{{(n - r)!r!}}. We will put the nCn^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 nCr^n{C_r}= n!(n−r)!r!\dfrac{{n!}}{{(n - r)!r!}}
Now we will put nCn^n{C_n} in the above formula to calculate its value.
nCn^n{C_n}= n!(n−n)!n!=n!0!n!=1\dfrac{{n!}}{{(n - n)!n!}} = \dfrac{{n!}}{{0!n!}} = 1
Hence, the value of nCn^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 nCr^n{C_r}= n!(n−r)!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 combinations5C2^5{C_2} = 5!(5−2)!2!=5!3!2!=5×4×3!3!×2×1=10\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.