Prabhakar went to buy masks, gloves and sanitizer. He wants to purchase total of 10items from amongst all such that he has one item of each type and he does not intend to purchase more than 5 masks. In how many ways can he select the items.

We need to determine how many ways Prabhakar can purchase 10 items, including masks, gloves, and sanitizers, given that:

1. He buys at least 1 item of each type (masks, gloves, and sanitizer).
2. He buys no more than 5 masks.
3. The total number of items he buys is exactly 10.

Let the number of masks, gloves, and sanitizers Prabhakar buys be denoted by \( x_1 \), \( x_2 \), and \( x_3 \), respectively, where:

- \( x_1 \) represents the number of masks,
- \( x_2 \) represents the number of gloves,
- \( x_3 \) represents the number of sanitizers.

The conditions are:
\[
x_1 + x_2 + x_3 = 10
\]
with the constraints:
\[
1 \leq x_1 \leq 5, \quad x_2 \geq 1, \quad x_3 \geq 1.
\]

### Step 1: Substituting for the lower bounds

Since Prabhakar buys at least 1 of each item, we subtract 1 from each of \( x_1 \), \( x_2 \), and \( x_3 \) to make a substitution:
\[
x_1' = x_1 - 1, \quad x_2' = x_2 - 1, \quad x_3' = x_3 - 1.
\]
Thus, \( x_1' \geq 0 \), \( x_2' \geq 0 \), and \( x_3' \geq 0 \). The equation becomes:
\[
(x_1' + 1) + (x_2' + 1) + (x_3' + 1) = 10,
\]
which simplifies to:
\[
x_1' + x_2' + x_3' = 7.
\]

Now, \( x_1' \) can take values from 0 to 4 (since \( 1 \leq x_1 \leq 5 \), implying \( 0 \leq x_1' \leq 4 \)), while \( x_2' \geq 0 \) and \( x_3' \geq 0 \).

### Step 2: Counting the solutions

We now need to count the number of non-negative integer solutions to the equation:
\[
x_1' + x_2' + x_3' = 7,
\]
where \( 0 \leq x_1' \leq 4 \) and \( x_2' \geq 0 \), \( x_3' \geq 0 \).

#### Case 1: \( x_1' = 0 \)
In this case, we have \( x_2' + x_3' = 7 \). The number of non-negative integer solutions to this equation is:
\[
\binom{7 + 2 - 1}{2 - 1} = \binom{8}{1} = 8.
\]

#### Case 2: \( x_1' = 1 \)
In this case, we have \( x_2' + x_3' = 6 \). The number of non-negative integer solutions is:
\[
\binom{6 + 2 - 1}{2 - 1} = \binom{7}{1} = 7.
\]

#### Case 3: \( x_1' = 2 \)
In this case, we have \( x_2' + x_3' = 5 \). The number of non-negative integer solutions is:
\[
\binom{5 + 2 - 1}{2 - 1} = \binom{6}{1} = 6.
\]

#### Case 4: \( x_1' = 3 \)
In this case, we have \( x_2' + x_3' = 4 \). The number of non-negative integer solutions is:
\[
\binom{4 + 2 - 1}{2 - 1} = \binom{5}{1} = 5.
\]

#### Case 5: \( x_1' = 4 \)
In this case, we have \( x_2' + x_3' = 3 \). The number of non-negative integer solutions is:
\[
\binom{3 + 2 - 1}{2 - 1} = \binom{4}{1} = 4.
\]

### Step 3: Adding up the cases

Now, we sum the number of solutions from each case:
\[
8 + 7 + 6 + 5 + 4 = 30.
\]

Thus, the total number of ways Prabhakar can select the items is \( \boxed{30} \).

---

Would you like more details on any of these steps or related topics?

Here are 5 additional questions to think about:
1. What would change if Prabhakar could buy a maximum of 6 masks instead of 5?
2. How would the solution change if gloves and sanitizer had upper limits on how many could be purchased?
3. What if Prabhakar was limited to purchasing exactly 3 sanitizers?
4. How can the stars and bars method be applied in other combinatorial problems?
5. What would happen if the total number of items to be bought was increased to 15?

**Tip**: When working on problems with constraints, always try to simplify the equation by removing the minimum values first to make calculations easier.

Prabhakar went to buy masks, gloves and sanitizer. He wants to purchase total of 10 items from amongst all such that he has one item of each type and he does not intend to purchase more than 5 masks. In how many ways can he select the items?

Solve the combinatorics problem where Prabhakar selects 10 items from masks, gloves, and sanitizers, with a maximum of 5 masks. This problem uses key mathematical concepts like the Stars and Bars Theorem and non-negative integer solutions. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation