Question

Number of ways of selecting 5 things from 10 similar things is:
A.1
B.$^{10}{C_5}$
C. $^9{C_5}$
D. $\dfrac{{^{10}{C_5}}}{{5!}}$

Answer 1

Hint: When we have to find the number of combination of some things from the given number of things then we will use the formula of combination,
$^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}$

Complete step-by-step answer:
Here total number of things is 10
Number of things which are chosen or combination are 5
The number of combination take place are carried out by $^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}$
Where, we will substitute the value of the total number of objects in place of n.
The number of choosing objects for combination is placed in place of r and the formula gives the total number of ways of combination.
After substituting the values we will get,
$
\Rightarrow ^{10}{C_5} = \dfrac{{10!}}{{5!\left( {10 - 5} \right)!}}\\
 = \dfrac{{10!}}{{5!5!}}
$
But according to question, here all the given ten things are similar then 10 will be replaced by 1 so we get,
$\Rightarrow$ $^1{C_5} = \dfrac{{1!}}{{5!\left( {1 - 5} \right)!}}$
If the given all 10 things are similar then selecting 5 things from them is also similar, so again replacing 5 with 1 then we will get,
$
\Rightarrow ^1{C_1} = \dfrac{{1!}}{{1!\left( {1 - 1} \right)!}}\\
\Rightarrow ^1{C_1} = \dfrac{{1!}}{{1!0!}}
$
The factorial value of 0 is 1 so calculating the values then we get,
$
\Rightarrow ^1{C_1} = \dfrac{1}{{1 \times 1}}\\
 = 1
$
So, when we select 5 things from 10 similar things then there should be only one combination because if we replace the things after that the combination remains the same as all are similar. Therefore, there should be only one possible combination. Hence option A is correct.

So, the correct answer is “Option A”.

Note: In this type of problem, if you will find that the total numbers of things are similar then you can directly give the number of solutions as 1 whether there are selecting things is 1 or more. As similar things always make one combination.