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Zorluk: OrtaPermutations, Combinations, and Fundamental Counting Principle

A financial firm has a pool of 1010 analysts, consisting of 66 senior analysts and 44 junior analysts. Which of the following selection procedures will yield EXACTLY 120120 unique possible groups? Select all such procedures.

  1. Forming a 55-member committee that contains exactly 33 senior analysts and 22 junior analystsCevap
  2. Forming a 33-member subcommittee from the entire pool of 1010 analysts without any restrictionsCevap
  3. Forming a 77-member project panel from the entire pool of 1010 analysts without any restrictionsCevap
  4. D
    Forming a 55-member committee that contains exactly 44 senior analysts and 11 junior analyst
  5. E
    Forming a 44-member task force from the entire pool of 1010 analysts without any restrictions

Cevap

The procedures that yield exactly 120 unique possible groups are: forming a 5-member committee with 3 senior and 2 junior analysts, forming a 3-member subcommittee from all 10 analysts, and forming a 7-member project panel from all 10 analysts.
The correct procedures are those that evaluate to exactly 120 combinations: (1) Selecting 3 senior analysts from 6 and 2 junior analysts from 4 gives \(\binom{6}{3} \times \binom{4}{2} = 20 \times 6 = 120\). (2) Choosing 3 analysts from 10 gives \(\binom{10}{3} = 120\). (3) Choosing 7 analysts from 10 is symmetric to choosing 3 analysts, yielding \(\binom{10}{7} = \binom{10}{3} = 120\).

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1
Calculate combinations for forming a 5-member committee with 3 senior and 2 junior analysts
\(\binom{6}{3} \times \binom{4}{2} = 20 \times 6 = 120\)
The selection of senior and junior analysts are independent decisions, so their combination values are multiplied together.
2
Calculate combinations for choosing 3 analysts out of 10 without restrictions
\(\binom{10}{3} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120\)
Order of selection does not matter, so the standard combination formula \(\binom{n}{k}\) is applied.
3
Calculate combinations for choosing 7 analysts out of 10 without restrictions
\(\binom{10}{7} = \binom{10}{10-7} = \binom{10}{3} = 120\)
Choosing 7 people to include is mathematically equivalent to choosing 3 people to exclude.
4
Evaluate the remaining options to verify they do not yield 120
\(\binom{6}{4} \times \binom{4}{1} = 60\) and \(\binom{10}{4} = 210\)
Neither of these evaluations equals the target value of 120.

Anahtar Kavram

Combinations and the Fundamental Counting Principle
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Permutations, Combinations, and Fundamental Counting Principle

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