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(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.
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(a) $2 \mathrm{M}, 5 \mathrm{P}, 3 \mathrm{C}$.
(i) Tie same books together
Maths books can be arranged within themselves in 2! ways, Physics books can be arranged in $5 !$ ways, Chemistry books in $3 !$ ways.
The three sets of subjects can be arranged in $3 !$ ways.
$\therefore$ Total arrangements $=3 ! 2 ! 5 ! 3 !$
$=6 \times 2 \times 120 \times 6=8640$ ways.
(ii) Tie Physics books together.
There are $1+2+3=6$ books to be arranged in $6 !$ ways.
The Physics books can be arranged in $5 !$ ways.
$\therefore$ Total arrangement $=6 ! 5 !=720 \times 120=86,400$ ways
(b) $p\left(\right.$ English speaking) $=p=\frac{13}{20} ; p($ non- English speaking $)=q=\frac{7}{20}$
8 persons selected,
$(p+q)^{8}=p^{8}+8 p^{7} q+28 p^{6} q^{2}+56 p^{5} q^{3}+70 p^{4} q^{4}+56 p^{3} q^{5}+28 p^{2} q^{6}+8 p q^{7}+q^{8}$
$p$ (at least 3 speak English) $=1-$ [none $+$ one $+t w o]$.
$=1-\left[q^{8}+8 p q^{7}+28 p^{2} q^{6}\right]$

\begin{aligned}
&=1-\left[(0.35)^{8}+8(0.65)(0.35)^{2}+28(0.65)^{2}(0.35)^{2}\right] \\
&=1-[0.0002252+0.003346+0.1131] \\
&=1-0.001488=0.98512 \\
&\cong 0.985 \text { (3 sig. figs). }
\end{aligned}
by Diamond (89,304 points)

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