Question

The total number of ways of selecting six coins out of 20 one-rupee coins, 10 fifty-paise coins and 7 twenty-five coins is:
A. 28
B. 56
C. $^{37}{C_6}$
D. None of these

Answer 1

Hint: Determine the number of types of coins from which we have to select 6 coins. There are 3 types of coins but there are more than one coin of each type, hence each of the type is repeating. The number of combinations of $r$ coins can be selected from $n$ types of coins, where repetition is allowed can be calculated as, $^{n + r - 1}{C_r}$.

Complete step-by-step answer:
We know that the type of coins are three, one-rupee coins, fifty-paise coins and twenty-five paise coins.
We have to select 6 coins from the total coins present.
The total types of coins can be represented as $n$, where $n = 3$.
And we have to select 6 coins, so, $r = 6$.
Since, there are more than one coin in each type, we will use the formula of combination in which repetition is allowed.
We can calculate the number of ways of selecting $r$ coins from $n$ types of coins by using the formula, $^{n + r - 1}{C_r}$
On substituting $n = 3$ and $r = 6$ in the above formula, we get,
$^{3 + 6 - 1}{C_6}{ = ^8}{C_6}$
The combination can be solved using the formula, $^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}$
Therefore,
$^8{C_6} = \dfrac{{8!}}{{6!\left( {8 - 6} \right)!}} \\\ = \dfrac{{8!}}{{6!2!}} \\\ = \dfrac{{8.7.6!}}{{6!2.1}} \\\ = \dfrac{{8.7}}{{2.1}} \\\ = \dfrac{{56}}{2} \\\ = 28 \\\$
Therefore, the total number of ways of selecting six coins out of 20 one-rupee coins, 10 fifty-paise coins and 7 twenty-five coins is 28.
Hence, option A is correct .

Note: Many students use the formula $^n{C_r}$ for calculating the total number of ways of selecting six coins from total 37 coins, which is incorrect because this formula is used only when repetition of objects is not allowed. But in this question, each of the coins is repeating, hence the required answer will be calculated using the formula, $^{n + r - 1}{C_r}$.