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Difficulty: Very hardCombinations and Group Selections

A project committee of 44 members is to be selected from a pool of nn senior engineers and 66 junior engineers, where n≥4n \ge 4. The committee must contain at least one senior engineer and at least one junior engineer. If there are exactly 310310 different possible ways to select the committee, what is the value of nn?

  1. A
    4
  2. 5Answer
  3. C
    6
  4. D
    7
  5. E
    8

Answer

The value of nn is 5.
Using complementary counting, the total number of ways to pick any 4 members from the (n+6)(n+6) total engineers is (n+64)\binom{n+6}{4}. The condition requires at least one senior engineer and at least one junior engineer. The invalid cases are when all 4 are junior engineers (which can occur in (64)=15\binom{6}{4} = 15 ways) or all 4 are senior engineers (which can occur in (n4)\binom{n}{4} ways). Setting up the equation: (n+64)−(64)−(n4)=310\binom{n+6}{4} - \binom{6}{4} - \binom{n}{4} = 310, which simplifies to (n+64)−(n4)=325\binom{n+6}{4} - \binom{n}{4} = 325. Testing n=5n = 5 gives (114)−(54)=330−5=325\binom{11}{4} - \binom{5}{4} = 330 - 5 = 325, which matches the given condition.

Step-by-Step Solution

1
Set up the combination formula for total unrestricted selections
Total selections from (n+6)(n + 6) engineers choosing 4 is (n+64)\binom{n+6}{4}.
Order of selection does not matter when forming a committee.
2
Identify the restricted (invalid) committee configurations
Committees with 0 senior engineers: (64)=15\binom{6}{4} = 15. Committees with 0 junior engineers: (n4)\binom{n}{4}.
The committee must contain at least one member from each category.
3
Formulate the equation using complementary counting
(n+64)−(n4)−15=310  ⟹  (n+64)−(n4)=325\binom{n+6}{4} - \binom{n}{4} - 15 = 310 \implies \binom{n+6}{4} - \binom{n}{4} = 325.
Subtracting invalid configurations from total configurations yields the valid configurations.
4
Test values for nn
For n=5n = 5: (114)−(54)=330−5=325\binom{11}{4} - \binom{5}{4} = 330 - 5 = 325.
Since 325=325325 = 325, n=5n = 5 satisfies the given constraint exactly.

Key Concept

Group selections with constraints using complementary counting: Valid=Total−Restricted\text{Valid} = \text{Total} - \text{Restricted}.
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Topics

Combinations and Group Selections

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