10 classmates could not agree on who would stand in the group photo along with the teacher for the yearbook. How many possible groups can be made such that there is at least one student with the teacher in the photo?
Question
10 classmates could not agree on who would stand in the group photo along with the teacher for the yearbook. How many possible groups can be made such that there is at least one student with the teacher in the photo?
Solution
The problem can be solved using the concept of combinations in mathematics.
The total number of ways to form a group with at least one student is the sum of the number of ways to form a group with exactly one student, exactly two students, and so on up to exactly ten students.
This can be calculated using the formula for combinations, which is nCr = n! / [(n-r)! * r!], where n is the total number of items, r is the number of items to choose, and "!" denotes factorial.
- For one student, the number of combinations is 10C1 = 10.
- For two students, the number of combinations is 10C2 = 45.
- For three students, the number of combinations is 10C3 = 120.
- For four students, the number of combinations is 10C4 = 210.
- For five students, the number of combinations is 10C5 = 252.
- For six students, the number of combinations is 10C6 = 210.
- For seven students, the number of combinations is 10C7 = 120.
- For eight students, the number of combinations is 10C8 = 45.
- For nine students, the number of combinations is 10C9 = 10.
- For all ten students, the number of combinations is 10C10 = 1.
Adding these up, the total number of possible groups is 10 + 45 + 120 + 210 + 252 + 210 + 120 + 45 + 10 + 1 = 1023.
So, there are 1023 possible groups that can be made such that there is at least one student with the teacher in the photo.
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