Question

The least value of the natural number $n$ satisfying $C(n,5) + C(n,6) > C(n + 1,5)$ is ?
A.$11$
B.$10$
C.$12$
D.$13$

Answer 1

Hint: In this question we have been given an expression
$C(n,5) + C(n,6) > C(n + 1,5)$ .
We will first write the expression in standard form by breaking the brackets i.e.
$^n{C_5}{ + ^n}{C_6}{ > ^{n + 1}}{C_5}$ .
We will use this formula to solve this question:
$^n{C_{r - 1}}{ + ^n}{C_r}{ = ^{n + 1}}{C_r}$ .

Complete step-by-step answer:
According to question we have:
$C(n,5) + C(n,6) > C(n + 1,5)$ .
We can write the above as:
$^n{C_5}{ + ^n}{C_6}{ > ^{n + 1}}{C_5}$
Now we will use this formula
$^n{C_{r - 1}}{ + ^n}{C_r}{ = ^{n + 1}}{C_r}$ .
From comparing we can write the expression as:
 $^n{C_{6 - 1}}{ + ^n}{C_6}$
Here we have
 $r = 6$
So by applying the formula we can write that
$^n{C_{6 - 1}}{ + ^n}{C_6}{ = ^{n + 1}}{C_6}$
We will put this value in the expression, so we have :
$^{n + 1}{C_6}{ > ^{n + 1}}{C_5}$
We can write the above expression also as :
$^{n + 1}{C_{n + 1 - 6}}{ > ^{n + 1}}{C_5}$
We can see that the powers are same , so we will eliminate all the common factors from above, it gives us:
$n + 1 - 6 > 5$
We will now solve this equation:
$n - 5 > 5$
It gives us
$n > 5 + 5 \Rightarrow n > 10$
We have to find the least value of $n$, which is greater than $10$
Now from the above given options, we can see that the least value which is greater than $10$ is
$11$ .
Hence the correct option is (a) $11$
So, the correct answer is “Option B”.

Note: We should note that in the above solution, we have used this property i.e.
$^n{C_r}{ = ^n}{C_{n - r}}$ .
So in this expression $^{n + 1}{C_6}$ by comparing, here in place of $n$ we have
$n + 1$
And,
 $r = 6$ .
So applying this property, we can write
$^{n + 1}{C_6}{ = ^{n + 1}}{C_{n + 1 - 6}}$