Answer

Verified

448.2k+ views

**Hint:**In order to solve this question, we have to know about what $^{n}{{C}_{r}}$ stands for and then substitute $^{n}{{C}_{r}}=\dfrac{n!}{r!(n-r)!}$ in the equation and then solve it. After solving, rearrange the value that we got according to the options and get the answer.

**Complete step by step answer:**

We have to find the value of $^{n}{{C}_{r}}{{+}^{n}}{{C}_{r-1}}$.

The meaning of $^{n}{{C}_{r}}$ in terms of permutation and combination is picking $r$ objects out of $n$ objects.

And in mathematical way $^{n}{{C}_{r}}$ can be represented as,

${{\therefore }^{n}}{{C}_{r}}=\dfrac{n!}{r!(n-r)!}..........................(1)$

Similarly, $^{n}{{C}_{r-1}}$ can also be written as,

${{\Rightarrow }^{n}}{{C}_{r-1}}=\dfrac{n!}{(r-1)!(n-r+1)!}...............(2)$

Also, we know that

$\Rightarrow n!=n\times (n-1)!..................(3)$

Now, from the question, we have

${{\therefore }^{n}}{{C}_{r}}{{+}^{n}}{{C}_{r-1}}$

Substituting values from equation (1) and (2), we get

$\Rightarrow \dfrac{n!}{r!(n-r)!}+\dfrac{n!}{(r-1)!(n-r+1)!}$

Taking $\dfrac{n!}{(r-1)!(n-r)!}$ common from both terms, we get

$\Rightarrow \dfrac{n!}{(r-1)!(n-r)!}\left( \dfrac{1}{r}+\dfrac{1}{n-r+1} \right)$

Simplifying the above equation, we get

$\Rightarrow \dfrac{n!}{(r-1)!(n-r)!}\left( \dfrac{n-r+1+r}{r(n-r+1)} \right)$

Cancelling out $-r$ with $r$, we get

$\Rightarrow \dfrac{n!}{(r-1)!(n-r)!}\times \dfrac{n+1}{r(n-r+1)}$

Rearranging the terms like multiplying $(n+1)$ with $n!$, $r$ with $(r-1)!$ and $(n-r+1)$ with $(n-r)!$ and using (3), we get

$\Rightarrow \dfrac{(n+1)!}{(r)!(n-r+1)!}=\dfrac{(n+1)!}{(r)!(n+1-r)!}$

Also the above expression can be written as

${{\Rightarrow }^{(n+1)}}{{C}_{r}}$

So, we get the value of $^{n}{{C}_{r}}{{+}^{n}}{{C}_{r-1}}{{=}^{(n+1)}}{{C}_{r}}$.

**So, the correct answer is “Option B”.**

**Note:**This question tests the understanding of expression solving and rearranging knowledge of students. In these type of question students often do two mistakes, first one is that they put the value of $^{n}{{C}_{r}}$ and other terms in factorial and then solve it but they get struck at the last step i.e. converting back the final expression in $^{n}{{C}_{r}}$, so they leave it. Second mistake it that they get confused in $^{n}{{C}_{r}}$ and \[^{n}{{P}_{r}}\], they mix these two things and end up getting wrong answer. So, carefully read the question and substitute the correct value and then rearrange the final expression to get the correct answer.

Recently Updated Pages

How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE

Mark and label the given geoinformation on the outline class 11 social science CBSE

When people say No pun intended what does that mea class 8 english CBSE

Name the states which share their boundary with Indias class 9 social science CBSE

Give an account of the Northern Plains of India class 9 social science CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Trending doubts

Which are the Top 10 Largest Countries of the World?

In Indian rupees 1 trillion is equal to how many c class 8 maths CBSE

How do you graph the function fx 4x class 9 maths CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

What organs are located on the left side of your body class 11 biology CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Difference Between Plant Cell and Animal Cell