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Subjects
Class 11MathematicsPermutations and CombinationsFundamental principle of countingHard
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A survey has (6) questions. The first (2) questions have (4) options each, the next (3) have (5) options each, and the last has (3) options. If option (D) is chosen in the first question, only (2) options remain in the last question. How many answer patterns are possible?

Answer and explanation

Correct answer: (5500)

The choices for different questions are combined by multiplication, but the special condition depends on whether the first answer is D. Therefore, split the counting into two disjoint cases. If the first answer is D, it has 1 choice, the second question has 4 choices, the next three questions have \(5^3\) choices, and the last question has only 2 choices. This gives \(1\times4\times5^3\times2=1000\).

If the first answer is not D, it has 3 choices. The second question still has 4 choices, each of the next three has 5 choices, and the last question then has all 3 choices. This gives \(3\times4\times5^3\times3=4500\). The cases cannot overlap, so add them: \(1000+4500=5500\). Hence option B is correct.

Related tags

Permutations-And-CombinationsFundamental-PrincipleCountingHard
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