Answer

Verified

447.9k+ views

**Hint:**We will use the property of the combinations which states that If we are given a condition that ${}^n{C_p} = {}^n{C_q}$, then any of these two situations will follow: (i): p = q and (ii): p + q = n to solve the given question. Here, n is the total number of ways to do a certain action then p or q are the individual ways to do it in a specific manner.

**Complete step-by-step answer:**We are given a condition that ${}^n{C_4} = {}^n{C_6}$.

Then we know that we have a property of combinations which states that if ${}^n{C_p} = {}^n{C_q}$, then either p = q or p + q = n.

Here, we have ${}^n{C_4} = {}^n{C_6}$, we can say that here p = 4 and q = 6.

Using the above property in the given condition, we get

Either of the two conditions satisfy i. e.,

(i): p = q

$ \Rightarrow 4$should be equal to 6 but $4 \ne 6$.

Hence, we can say that this situation is not possible.

Looking at the first situation, we get that second situation must follow as the first is not possible.

(ii): p + q = n

$

\Rightarrow 4 + 6 = n \\

\Rightarrow 10 = n \\

$

Hence, substituting this value of n in ${}^{12}{C_n}$, we get

$ \Rightarrow {}^{12}{C_{10}}$ is the required value we need to calculate.

Upon simplification using the formula ${}^n{C_r} = \dfrac{{n!}}{{\left( {n - r} \right)!r!}}$ , we get

$ \Rightarrow {}^{12}{C_{10}} = \dfrac{{12!}}{{\left( {12 - 10} \right)!10!}}$

$ \Rightarrow {}^{12}{C_{10}} = \dfrac{{12!}}{{2!10!}}$

$ \Rightarrow {}^{12}{C_{10}} = \dfrac{{12 \times 11}}{{2 \times 1}} = 66$

Therefore, we can say that the value $ \Rightarrow {}^{12}{C_{10}}$is 66.

**Note:**This problem is not tough but tricky. We can also find the required value by evaluating the given condition ${}^n{C_4} = {}^n{C_6}$ for the value of n and then substituting it in $ \Rightarrow {}^{12}{C_{10}}$.

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

Difference Between Plant Cell and Animal Cell

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

Which are the Top 10 Largest Countries of the World?

Give 10 examples for herbs , shrubs , climbers , creepers

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

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

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

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

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