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Difficulty: MediumCounting Principles

A logistics officer needs to form a specialized transport convoy consisting of exactly 22 cargo trucks and 11 security vehicle. If there are 66 available cargo trucks and 33 available security vehicles, how many different combinations of vehicles can be selected for the convoy?

  1. A
    18
  2. 45Answer
  3. C
    84
  4. D
    90

Answer

45
The total number of convoy combinations is found by calculating the combinations for each vehicle type and multiplying them together. Choosing 22 cargo trucks from 66 yields 1515 combinations, and choosing 11 security vehicle from 33 yields 33 combinations. Multiplying these results gives 4545 possible convoys.

Step-by-Step Solution

1
Calculate the number of ways to select the cargo trucks.
6C2=6!2!(6−2)!=15_{6}C_{2} = \frac{6!}{2!(6-2)!} = 15
Since the selection order does not matter, use combinations to choose 22 cargo trucks from the 66 available.
2
Calculate the number of ways to select the security vehicle.
3C1=3_{3}C_{1} = 3
Use combinations to choose 11 security vehicle from the 33 available.
3
Determine the total combinations for the convoy.
15×3=4515 \times 3 = 45
Apply the Fundamental Counting Principle by multiplying the independent category selections together.

Key Concept

Combinations and the Fundamental Counting Principle
Estimated Time:1m 30s
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Topics

Counting Principles

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