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At a deli counter, there are 7 sandwiches with meat and 10 vegetarian sandwiches. Milan is at the counter buying sandwiches for a picnic. In how many ways can he choose 8 sandwiches if more than 5 must be vegetarian sandwiches?

At a deli counter, there are 7 sandwiches with meat and 10 vegetarian sandwiches. Milan is at the counter buying sandwiches for a picnic. In how many ways can he choose 8 sandwiches if more than 5 must be vegetarian sandwiches?

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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**Combinatorics and Probability Problems: Choosing Sandwiches** *Problem Statement:* At a deli counter, there are 7 sandwiches with meat and 10 vegetarian sandwiches. Milan is at the counter buying sandwiches for a picnic. In how many ways can he choose 8 sandwiches if more than 5 must be vegetarian sandwiches? *Solution Explanation:* 1. **Initialize the Problem**: - Meat sandwiches available: 7 - Vegetarian sandwiches available: 10 - Total sandwiches to choose: 8 - Minimum vegetarian sandwiches required: 6 2. **Cases to Consider**: Milan needs to choose at least 6 vegetarian sandwiches. We will consider how many ways he can select different combinations of sandwiches in the following cases: - Case 1: 6 vegetarian and 2 meat sandwiches - Case 2: 7 vegetarian and 1 meat sandwich - Case 3: 8 vegetarian sandwiches 3. **Mathematical Computation**: - **Case 1**: Choosing 6 vegetarian and 2 meat sandwiches - Ways to choose 6 vegetarian sandwiches from 10: \( \binom{10}{6} \) - Ways to choose 2 meat sandwiches from 7: \( \binom{7}{2} \) - Combine these: \( \binom{10}{6} \times \(\binom{7}{2} \) - **Case 2**: Choosing 7 vegetarian and 1 meat sandwich - Ways to choose 7 vegetarian sandwiches from 10: \( \binom{10}{7} \) - Ways to choose 1 meat sandwich from 7: \( \binom{7}{1} \) - Combine these: \( \binom{10}{7} \times \(\binom{7}{1} \) - **Case 3**: Choosing 8 vegetarian sandwiches - Ways to choose 8 vegetarian sandwiches from 10: \( \binom{10}{8} \) 4. **Combine All Cases** to get the total number of ways: \(\binom{10}{6} \times \binom{7}{2} + \binom{10}{7} \times \binom{7}{1} + \(\binom{10}{8} \) *Interactive
Transcribed Image Text:**Combinatorics and Probability Problems: Choosing Sandwiches** *Problem Statement:* At a deli counter, there are 7 sandwiches with meat and 10 vegetarian sandwiches. Milan is at the counter buying sandwiches for a picnic. In how many ways can he choose 8 sandwiches if more than 5 must be vegetarian sandwiches? *Solution Explanation:* 1. **Initialize the Problem**: - Meat sandwiches available: 7 - Vegetarian sandwiches available: 10 - Total sandwiches to choose: 8 - Minimum vegetarian sandwiches required: 6 2. **Cases to Consider**: Milan needs to choose at least 6 vegetarian sandwiches. We will consider how many ways he can select different combinations of sandwiches in the following cases: - Case 1: 6 vegetarian and 2 meat sandwiches - Case 2: 7 vegetarian and 1 meat sandwich - Case 3: 8 vegetarian sandwiches 3. **Mathematical Computation**: - **Case 1**: Choosing 6 vegetarian and 2 meat sandwiches - Ways to choose 6 vegetarian sandwiches from 10: \( \binom{10}{6} \) - Ways to choose 2 meat sandwiches from 7: \( \binom{7}{2} \) - Combine these: \( \binom{10}{6} \times \(\binom{7}{2} \) - **Case 2**: Choosing 7 vegetarian and 1 meat sandwich - Ways to choose 7 vegetarian sandwiches from 10: \( \binom{10}{7} \) - Ways to choose 1 meat sandwich from 7: \( \binom{7}{1} \) - Combine these: \( \binom{10}{7} \times \(\binom{7}{1} \) - **Case 3**: Choosing 8 vegetarian sandwiches - Ways to choose 8 vegetarian sandwiches from 10: \( \binom{10}{8} \) 4. **Combine All Cases** to get the total number of ways: \(\binom{10}{6} \times \binom{7}{2} + \binom{10}{7} \times \binom{7}{1} + \(\binom{10}{8} \) *Interactive
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