In how many ways can Rs. 25 be paid to a person using at least one denomination each of Re. 1, Rs. 2 and Rs. 5?

To solve the problem of determining the number of ways Rs. 25 can be paid using at least one denomination each of Re. 1, Rs. 2, and Rs. 5, we can use the following steps:

### Step 1: Define Variables
Let:
- \( x_1 \) be the number of Re. 1 coins.
- \( x_2 \) be the number of Rs. 2 coins.
- \( x_5 \) be the number of Rs. 5 coins.

The equation representing the total amount is:
\[
x_1 + 2x_2 + 5x_5 = 25
\]
Where \( x_1 \geq 1 \), \( x_2 \geq 1 \), and \( x_5 \geq 1 \) since we must use at least one of each denomination.

### Step 2: Subtract the Minimum Required Coins
To simplify, subtract 1 from each \( x_1 \), \( x_2 \), and \( x_5 \) to account for the minimum coins:

Let:
\[
y_1 = x_1 - 1, \quad y_2 = x_2 - 1, \quad y_5 = x_5 - 1
\]
So now, \( y_1, y_2, y_5 \geq 0 \), and the equation becomes:
\[
(y_1 + 1) + 2(y_2 + 1) + 5(y_5 + 1) = 25
\]
Simplifying:
\[
y_1 + 2y_2 + 5y_5 = 17
\]
Now, we need to find the number of non-negative integer solutions to the equation \( y_1 + 2y_2 + 5y_5 = 17 \).

### Step 3: Counting Solutions
We can solve this by iterating over possible values of \( y_5 \) and finding corresponding solutions for \( y_1 \) and \( y_2 \).

- \( y_5 = 0 \): \( y_1 + 2y_2 = 17 \) (Possible \( y_2 \) values: \( 0, 1, 2, \ldots, 8 \))
  - Number of solutions: 9
- \( y_5 = 1 \): \( y_1 + 2y_2 = 12 \) (Possible \( y_2 \) values: \( 0, 1, 2, \ldots, 6 \))
  - Number of solutions: 7
- \( y_5 = 2 \): \( y_1 + 2y_2 = 7 \) (Possible \( y_2 \) values: \( 0, 1, 2, 3 \))
  - Number of solutions: 4
- \( y_5 = 3 \): \( y_1 + 2y_2 = 2 \) (Possible \( y_2 \) values: \( 0, 1 \))
  - Number of solutions: 2

### Step 4: Add the Number of Solutions
Total number of solutions is:
\[
9 + 7 + 4 + 2 = 22
\]

Thus, there are **22 different ways** to pay Rs. 25 using at least one denomination each of Re. 1, Rs. 2, and Rs. 5.

Would you like more details or have any questions?

Here are some related questions:
1. How can you generalize this method to other amounts?
2. What would change if there was an additional Rs. 10 denomination?
3. How many ways can Rs. 50 be paid using at least one of Re. 1, Rs. 2, Rs. 5, and Rs. 10?
4. What if we restrict the number of coins for each denomination?
5. How can the stars and bars method be applied to such problems?

**Tip:** When solving combinatorial problems with constraints, transforming variables by subtracting the minimum requirement helps simplify the calculations.

Discover how many ways Rs. 25 can be paid using at least one denomination each of Re. 1, Rs. 2, and Rs. 5 coins. Learn the application of Diophantine equations and combinatorial methods in solving this problem. Suitable for Grades 7-9.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation