ITPEC IP October 2022 Question 2
There are five (5) lamps arranged in line. Each of them can be turned on and off independently. How many patterns are there where two (2) or more lamps are turned on? Here, the pattern where all lamps are turned on is counted as one (1) and the pattern where any one of them is turned on is counted as five (5).
Explanation
26 — count every on/off pattern of the five lamps, then remove those with fewer than two lamps on.
- •All patterns: each lamp is on or off independently, so 25 = 32
- •No lamp on: 1 pattern
- •Exactly one lamp on: 5 patterns, one for each lamp
- •Two or more lamps on: 32 − 1 − 5 = 26
Check by counting each case directly: C(5, 2) + C(5, 3) + C(5, 4) + C(5, 5) = 10 + 10 + 5 + 1 = 26.
Answer (c)
Why not others:
- •(a) 4 — counts how many lamps can be on (2, 3, 4 or 5), not which lamps: each of those numbers covers several patterns
- •(b) 10 — C(5, 2), only the patterns with exactly two lamps on; three, four and five lamps on also count
- •(d) 32 — 25, every pattern, including the one with all lamps off and the five with a single lamp on
Key rule: For "two or more", subtract the few excluded cases (none and exactly one) from the 2n total; remember to remove the all-off pattern too.
Basic Theory · AI-written, may contain errors
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