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Difficulty: MediumCombinations

A international conference delegation of 66 members is to be selected from 55 diplomats and 44 translators. In how many distinct ways can the delegation be formed if it must include at least 44 diplomats?

Answer: 34 ways

Answer

The total number of distinct ways to form the delegation is 34.
To form a delegation of 6 with at least 4 diplomats out of 5 diplomats and 4 translators, we evaluate two mutually exclusive cases: selecting 4 diplomats and 2 translators (5C4 * 4C2 = 30 ways) and selecting 5 diplomats and 1 translator (5C5 * 4C1 = 4 ways). Summing these gives 34 distinct ways.

Step-by-Step Solution

1
Determine the valid combinations of diplomats and translators.
Two valid cases exist: (4 diplomats, 2 translators) and (5 diplomats, 1 translator).
The total delegation size is 6 and it must contain at least 4 diplomats out of the 5 available.
2
Compute the selection ways for Case 1 (4 diplomats and 2 translators).
5 * 6 = 30 ways
Selecting 4 diplomats out of 5 is 5C4 = 5 ways, and selecting 2 translators out of 4 is 4C2 = 6 ways.
3
Compute the selection ways for Case 2 (5 diplomats and 1 translator).
1 * 4 = 4 ways
Selecting 5 diplomats out of 5 is 5C5 = 1 way, and selecting 1 translator out of 4 is 4C1 = 4 ways.
4
Sum the combinations from both mutually exclusive cases.
30 + 4 = 34 ways
By the addition principle of counting, mutually exclusive scenarios are added together.

Key Concept

Combinations with constraints and mutually exclusive cases
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Topics

Combinations

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