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A committee must be formed with 2 teachers and 3 students. If there are 7 teachers to choose from, and 17 students, how many different ways could the committee be made?
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A committee must be formed with 22 teachers and 33 students. If there are 77 teachers to choose from, and 1717 students, how many different ways could the committee be made?\newlineAnswer:

Full solution

Q. A committee must be formed with 22 teachers and 33 students. If there are 77 teachers to choose from, and 1717 students, how many different ways could the committee be made?\newlineAnswer:
  1. Calculate Teachers Combinations: To determine the number of different ways to form the committee, we need to calculate the combinations of teachers and students separately and then multiply them together. For the teachers, we need to choose 22 out of 77, which is a combination problem. The formula for combinations is C(n,k)=n!k!(nk)!C(n, k) = \frac{n!}{k!(n-k)!}, where nn is the total number to choose from, kk is the number to choose, and !! denotes factorial.
  2. Calculate Students Combinations: First, we calculate the number of ways to choose 22 teachers out of 77. Using the combination formula:\newlineC(7,2)=7!(2!(72)!)=7!(2!5!)=(7×6)(2×1)=422=21C(7, 2) = \frac{7!}{(2!(7-2)!)} = \frac{7!}{(2!5!)} = \frac{(7 \times 6)}{(2 \times 1)} = \frac{42}{2} = 21 ways to choose the teachers.
  3. Multiply Teachers and Students Combinations: Next, we calculate the number of ways to choose 33 students out of 1717. Using the combination formula:\newlineC(17,3)=17!3!(173)!=17!3!14!=17×16×153×2×1=40806=680C(17, 3) = \frac{17!}{3!(17-3)!} = \frac{17!}{3!14!} = \frac{17 \times 16 \times 15}{3 \times 2 \times 1} = \frac{4080}{6} = 680 ways to choose the students.
  4. Total Number of Committees: Finally, we multiply the number of ways to choose the teachers by the number of ways to choose the students to find the total number of different committees that can be formed.\newlineTotal number of committees == Number of ways to choose teachers ×\times Number of ways to choose students =21×680= 21 \times 680.
  5. Perform Multiplication: Now we perform the multiplication:\newline21×680=14,28021 \times 680 = 14,280.\newlineSo, there are 14,28014,280 different ways the committee can be formed.

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