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Difficulty: MediumCombinations and Group Selections

A cybersecurity firm must select a 4-tool diagnostic suite from a pool of 6 network scanners and 4 code analyzers. If the suite must contain at least 1 network scanner and at least 1 code analyzer, how many different 4-tool suites can be selected?

Answer: 194

Answer

The total number of different 4-tool suites that can be selected is 194.
The total number of ways to select any 4 tools from the pool of 10 is (104)=210\binom{10}{4} = 210. The only selections that violate the requirement of having at least one tool of each type are choosing 4 scanners and 0 analyzers ((64)=15\binom{6}{4} = 15 ways) or 0 scanners and 4 analyzers ((44)=1\binom{4}{4} = 1 way). Subtracting these 16 invalid outcomes from the total gives 210−16=194210 - 16 = 194 valid 4-tool suites.

Step-by-Step Solution

1
Calculate the total unconstrained selections
210
There are 6+4=106 + 4 = 10 total tools, and 4 must be chosen. (104)=10×9×8×74×3×2×1=210\binom{10}{4} = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1} = 210.
2
Determine the number of invalid combinations violating the constraint
16 invalid combinations
The constraint requires at least 1 scanner and 1 analyzer. Invalid suites consist entirely of scanners ((64)=15\binom{6}{4} = 15) or entirely of analyzers ((44)=1\binom{4}{4} = 1).
3
Apply complementary counting to find the valid combinations
194 valid combinations
Subtract the invalid combinations from total unconstrained combinations: 210−(15+1)=194210 - (15 + 1) = 194.

Key Concept

Combinations with Group Selections and Complementary Counting
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Topics

Combinations and Group Selections

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