How many different committees can be formed from 12 teachers and 43 students if the committee consists of 3 teachers and 4 ​students? T

Question

How many different committees can be formed from 12 teachers and 43 students if the committee consists of 3 teachers and 4 ​students?
The committee of 7 members can be selected in BLANK
different ways.​

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Eirian 4 years 2021-08-09T10:46:23+00:00 1 Answers 71 views 0

Answers ( )

  1. Neala
    0
    2021-08-09T10:47:52+00:00
    Reply

    Answer:

    \displaystyle 27150200

    Step-by-step explanation:

    we are two conditions

    • committees can be formed from 12 teachers and 43 students
    • the committee consists of 3 teachers and 4 students

    In choosing a committee, order doesn’t matter; in case of teachers we need the number of combinations of 3 people chosen from 12

    remember that,

    \displaystyle\binom{n}{r} = \frac{n!}{r!(n - r)!}

    with the condition we obtain that,

    • n = 12
    • r = 3

    therefore substitute:

    \displaystyle\binom{12}{3} = \frac{12!}{3!(12 - 3)!}

    simplify Parentheses:

    \displaystyle\binom{12}{3} = \frac{12!}{3! \cdot9!}

    rewrite:

    \rm \displaystyle\binom{12}{3} = \frac{12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{(1 \times 2 \times 3 )\cdot1 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \times 8 \times 9}

    reduce fraction:

    \rm \displaystyle\binom{12}{3} = \frac{12 \times 11 \times 10}{1 \times 2 \times 3 }

    rewrite 12 and 10:

    \rm \displaystyle\binom{12}{3} = \frac{3 \times 2 \times 2 \times 11 \times 10}{1 \times 2 \times 3 }

    reduce fraction:

    \rm \displaystyle\binom{12}{3} = 2 \times 11 \times 10

    simplify multiplication:

    \rm \displaystyle\binom{12}{3} = 220

    In case of students we need the number of combinations of 4 students choosen from 43 therefore,

    \displaystyle\binom{43}{4} = \frac{43!}{4!(43 - 4)!}

    simplify which yields:

    \displaystyle\binom{43}{4} = 123410

    hence,

    The committee of 7 members can be selected in BLANK different ways is

    \displaystyle 123410 \times 220

    \displaystyle \boxed{27150200}

    and we’re done!

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