📘 Quiz

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Question 1 / 3
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1
Determine the number of distinct arrangements possible using all characters in the sequence 'ALIVE'.
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Solution: Step 1: Count unique characters in sequence = 5 Step 2: Apply permutation formula for n unique items = 5! (5 factorial) Step 3: Calculate factorial: 5! = 5 * 4 * 3 * 2 * 1 = 120
2
Arrange the letters of the word 'SAMPLE' such that no three vowels are together. How many arrangements are possible?
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Solution: Step 1: Total letters = 6 (S, A, M, P, L, E) Step 2: Vowels = A, E Step 3: Treat vowels as separate entities with gaps: _ C _ C _ C _ (C = consonant) Step 4: Arrange 4 consonants in 4! ways = 24 Step 5: Place vowels in 5 gaps (no three together): C(5,2) = 10 Step 6: Total arrangements = 24 * 10 = 240
3
Arrange 13 distinct symbols containing 6 specific markers in 6 designated slots while freely arranging the remaining 7 symbols. How many valid arrangements exist?
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Solution: Step 1: Identify markers (6) and slots (6 even positions) Step 2: Arrange 6 markers in 6 slots: 6! / (2!*2!) = 180 (due to repeated markers) Step 3: Arrange 7 remaining symbols in 7 slots: 7! / 2! = 2520 (due to repeated symbols) Step 4: Total arrangements = 180 * 2520 = 453600
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