Ways to Contribute to a Charitable Fund Using a Variety of Coin Denominations

In this article, we explore the methodology to determine the number of different ways a person can contribute to a charitable fund using a specific set of coins in his possession. The coins are of various denominations, and we will break down the problem step-by-step.

Breakdown of Coins

The person has a total of 38 individual coins in his bag, consisting of differing denominations as follows: 15 coins of Rs. 10 10 coins of Rs. 5 5 coins of Rs. 2 8 coins of Rs. 1

Contribution Calculation

To determine the total number of different ways to contribute to the charitable fund, we need to consider each type of coin and the number of coins available for each denomination.

Rs. 10 Coins: There are 15 coins, and the person can choose 0 to 15 coins. The number of ways to choose is given by the formula n 1, where n is the number of coins. Therefore, the number of ways is 15 1 16. Rs. 5 Coins: There are 10 coins, and the person can choose 0 to 10 coins. The number of ways is 10 1 11. Rs. 2 Coins: There are 5 coins, and the person can choose 0 to 5 coins. The number of ways is 5 1 6. Rs. 1 Coins: There are 8 coins, and the person can choose 0 to 8 coins. The number of ways is 8 1 9.

To find the total number of different ways to contribute to the charitable fund, we multiply the number of ways for each type of coin:

total ways 16 x 11 x 6 x 9

Let's perform the calculation step-by-step:

16 x 11 176 176 x 6 1056 1056 x 9 9504

Therefore, the total number of different ways the person can contribute to the charitable fund is 9504.

Coins Distinct vs. Not Distinct

When considering if the coins are distinct or not, we have two scenarios to explore:

Coins Distinct

If every coin is distinct and can either be present or absent from the donation, the person can contribute in 238 ways. This is nearly three hundred billion ways.

Coins Not Distinct

If the coins of the same denomination are not distinct, we need to consider the combinations of each denomination:

Rs. 10 Coins: The person can contribute in 16 ways (0 to 15 coins). Rs. 5 Coins: The person can contribute in 11 ways (0 to 10 coins). Rs. 2 Coins: The person can contribute in 6 ways (0 to 5 coins). Rs. 1 Coins: The person can contribute in 9 ways (0 to 8 coins).

The total number of different ways to make a donation, including the option of zero amount, is given by:

16 x 11 x 6 x 9 9504

Amount to Be Given

Another question to consider is how many different total amounts can be given. The smallest amount is zero, and the largest is the sum of all the coins:

Total amount 15 x 10 10 x 5 5 x 2 8 x 1 218 rupees

To determine all possible amounts, we use combinations of the coins. Let's break this down step-by-step:

Using Rs. 10 and Rs. 5 Coins

In the range from 0 to 200, every multiple of 5 can be achieved with combinations of Rs. 10 and Rs. 5 coins. For amounts 201 to 218, we use one or more Rs. 1 and Rs. 2 coins to achieve the required totals. Therefore, all amounts from 0 to 218 rupees can be given, resulting in a total of 219 options.

Note

It is important to note that not all piles of coins can result in every possible value. For example, if the total is 218 using 17 Rs. 10 coins, 9 Rs. 5 coins, and one each of Rs. 1 and Rs. 2 coins, some intermediate values are impossible to achieve.