At a deli counter, there are 8 sandwiches with meat and 9 vegetarian sandwiches. Lena is at the counter buying sandwiches for a picnic. In how many ways can she choose 8 sandwiches if 7 or more must be vegetarian sandwiches?(If necessary, consult a list of formulas.)
Question
At a deli counter, there are 8 sandwiches with meat and 9 vegetarian sandwiches. Lena is at the counter buying sandwiches for a picnic. In how many ways can she choose 8 sandwiches if 7 or more must be vegetarian sandwiches?(If necessary, consult a list of formulas.)
Solution
To solve this problem, we need to use the concept of combinations from combinatorics. The problem can be divided into two cases:
Case 1: Lena chooses exactly 7 vegetarian sandwiches. Case 2: Lena chooses all 8 sandwiches as vegetarian.
Case 1: Lena chooses exactly 7 vegetarian sandwiches. This means she will choose 1 sandwich with meat. The number of ways she can do this is given by the combination formula C(n, k) = n! / [k!(n-k)!], where n is the total number of items, k is the number of items to choose, and '!' denotes factorial.
The number of ways to choose 7 vegetarian sandwiches from 9 is C(9, 7) = 9! / [7!(9-7)!] = 36. The number of ways to choose 1 meat sandwich from 8 is C(8, 1) = 8! / [1!(8-1)!] = 8.
So, the total number of ways for Case 1 is 36 * 8 = 288.
Case 2: Lena chooses all 8 sandwiches as vegetarian. The number of ways to choose 8 vegetarian sandwiches from 9 is C(9, 8) = 9! / [8!(9-8)!] = 9.
So, the total number of ways for Case 2 is 9.
Adding the results from both cases, the total number of ways Lena can choose 8 sandwiches if 7 or more must be vegetarian is 288 + 9 = 297.
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