(a) Two Mathematics books, 5 different Physics books and 3 different Chemistry are to be arranged on a shelf. How many arrangements are possible if ; (i) bo...
Question 1 Report
(a) Two Mathematics books, 5 different Physics books and 3 different Chemistry are to be arranged on a shelf. How many arrangements are possible if ;
(i) books on the same subject must stand together? (ii) only the Physics books must stand together?
(b) In a certain community, 13 out of every 20 persons speak English. If 8 persons are selected at random from the community, find, correct to three significant figures, the probability that at least 3 of them speak English.
Answer Details
(a)(i) If books of the same subject must stand together, treat each subject as one block: 3 blocks arranged in \(3!\) ways, with the books inside each block also arranged.
\[3! \times 2! \times 5! \times 3! = 6 \times 2 \times 120 \times 6 = 8640\](ii) If only the Physics books must stand together, treat the 5 Physics books as one block. The items to arrange are: 2 Maths + 1 Physics block + 3 Chemistry \(= 6\) items.
\[6! \times 5! = 720 \times 120 = 86400\](b) \(p = \tfrac{13}{20} = 0.65,\ q = 0.35,\ n = 8\). Find \(P(\text{at least 3})\) via the complement:
\[P(0) = (0.35)^8 = 0.000225\] \[P(1) = \binom{8}{1}(0.65)(0.35)^7 = 0.003346\] \[P(2) = \binom{8}{2}(0.65)^2(0.35)^6 = 0.021747\] \[P(\ge 3) = 1 - (0.000225 + 0.003346 + 0.021747) = 1 - 0.025318 = 0.975\]The probability that at least 3 speak English is \(0.975\) (3 s.f.).
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