Question

The total number of ways in which a beggar can be given at least one rupee from four 25 paise coins, three 50 paisa coins and 2 one rupee coins, is
(a) 54
(b) 53
(c) 51
(d) None of these

Answer 1

Hint: To solve the question given above, we will find the total number of ways in which we can select one or more from 1 rupee coins, 50 paise coins, and 25 paise coins. Then, we will subtract those cases from it in which we are giving 0 paise, 25 paise, 50 paise and 75 paise to the beggar.

Complete step by step solution:
In this question, we will first find out the total number of ways in which we can select none, one or more from four 25 paise coins, three 50 paise coins, and 2 one rupee coins. For this, we will use the following:
The total number in which we can select none, one or more from p – alike, q – alike, r – alike and N distinct objects is given by:
$\text{Total Number of ways }=\left( p+1 \right)\left( q+1 \right)\left( r+1 \right){{2}^{N}}$
In our case, the value of p is 4, the value of q is 3 and the value of r is 2. In our case, there are no distinct objects, so the value of N in our case will be 0. Thus, the total number of ways of giving the beggar any number of coins will be given by:
$\Rightarrow \text{Total Number of ways }=\left( 4+1 \right)\left( 3+1 \right)\left( 2+1 \right){{2}^{0}}$
$\Rightarrow \text{Total Number of ways }=5\times 4\times 3\times 1$
$\Rightarrow \text{Total Number of ways }=60$
The above found is not the answer because the minimum amount we have to give to the beggar is 1 rupee. But, in the total number of ways, there will be times when the beggar will be given 0 paise, 25 paise, 50 paise, and 75 paise. Thus, we will subtract these from the total number of ways.
Ways of giving 0 paise will be 1.
Ways of giving 25 paise will be 1 (One 25 paise coin can be given).
Ways of giving 50 paise will be 2 (Either he can be given two 25 rupee coins or he can be given one 50 paise coin).
Ways of giving 75 paise will be 2 (Either he can be given three 25 paise coins or he can be given one 25 paise and one 50 paise coins).
Thus, the required number of ways = Total number of ways – Ways in which he is given 0 paise, 25 paise, 50 paise and 75 paise.
Required ways = 60 – (1 + 1 + 2 + 2) = 60 – 6 = 54
Therefore, the required ways are 54.
Hence, option (a) is the right answer.

Note: While applying this formula, we have assumed that all the coins of the same quantity are similar to each other, i.e, one 25 paise coin is similar to another 25 paise coin, one 50 paise coin is similar to another 50 paise coin. The formula used can be applied if the coins are not identical but the answer will be different because in this case, N will not be equal to 0.