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Zorluk: OrtaCombinations and Group Selections

A task force must select a 4-member subcommittee from a pool consisting of 6 architects and nn urban planners. If there are exactly 111 different possible subcommittees that include at least 1 urban planner, what is the value of nn?

  1. A
    2
  2. 3Cevap
  3. C
    4
  4. D
    5
  5. E
    9

Cevap

The value of nn is 3.
Using the complementary counting method, the number of subcommittees with at least 1 urban planner equals the total possible 4-member subcommittees from the pool of (6+n)(6+n) people minus the subcommittees formed entirely of architects. The number of all-architect subcommittees is (64)=15\binom{6}{4} = 15. Adding this to 111 gives (6+n4)=126\binom{6+n}{4} = 126. Since (94)=126\binom{9}{4} = 126, the total pool size is 9 people. Subtracting the 6 architects gives n=3n = 3.

Adım Adım Çözüm

1
Apply the complementary counting principle
Number of valid subcommittees = (Total subcommittees) - (Subcommittees with 0 urban planners)
It is easier to subtract subcommittees containing only architects from the total number of subcommittees than to sum cases with 1, 2, 3, or 4 urban planners.
2
Calculate the number of subcommittees consisting only of architects
\binom{6}{4} = \frac{6 \times 5}{2 \times 1} = 15
Choosing 4 members from the 6 available architects gives 15 possible all-architect subcommittees.
3
Set up the combination equation for the total pool size
\binom{6+n}{4} - 15 = 111 \implies \binom{6+n}{4} = 126
Adding 15 to both sides isolates the combination expression representing selections from the total pool of 6+n6+n people.
4
Solve for the total pool size and isolate nn
\binom{9}{4} = \frac{9 \times 8 \times 7 \times 6}{4 \times 3 \times 2 \times 1} = 126 \implies 6+n = 9 \implies n = 3
Evaluating combinations shows that selecting 4 items from 9 gives 126. Subtracting the 6 architects leaves 3 urban planners.

Anahtar Kavram

Group Selection with Restrictions and Complementary Counting
Tahmini Süre:2m 0s
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Combinations and Group Selections

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