Question

a coin is tossed four times and the sequence of heads (h) and tails (t) is recorded.
(a) use the generalized multiplication principle to determine the number of outcomes of this activity.
□ outcomes
(b) find all possible sequences:
○ hhhh, hhht, hhth, hhtt, hthh, htht, htth, httt, thhh, thht, thth, thtt, tthh, ttht, ttth, and tttt.
○ hhhh, hhht, hhth, hhtt, hthh, htht, htth, httt, thhh, thht, thth, thtt, tthh, ttht, ttth, and hhhh.
○ hhh, hht, hth, htt, thh, tht, tth, and ttt.
○ hhhh, hhht, hhth, hhtt, hthh, htht, htth, and httt.
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two commuter trains and three express buses depart from city a to city b in the morning, and five commuter trains and five express buses operate on the return trip in the evening. in how many ways can a commuter from city a to city b complete a daily round trip via bus and/or train?
□ ways
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Explanation:

Step1: Calculate coin toss outcomes

Each toss has 2 results. For 4 tosses: $2 \times 2 \times 2 \times 2 = 2^4$

Step2: Evaluate the coin toss total

$2^4 = 16$

Step3: Morning travel options

Add trains and buses: $2 + 3 = 5$

Step4: Evening travel options

Add trains and buses: $5 + 5 = 10$

Step5: Calculate round trip ways

Multiply morning and evening options: $5 \times 10 = 50$

Step6: Verify coin sequences

List all 4-length H/T sequences (16 total)

Answer:

1. (a) 16

(b) HHHH, HHHT, HHTH, HHTT, HTHH, HTHT, HTTH, HTTT, THHH, THHT, THTH, THTT, TTHH, TTHT, TTTH, and TTTT

1. 50