(a) Edem and his wife were invited to a dinner by a family of 5. They all sat in such a way in such a way that Edem sat next to his wife. Find the number of...

Answer Details

(a) Edem and his wife join a family of 5, so 7 people sit in a row with Edem next to his wife.

Tie Edem and his wife into a single block. Then there are \(6\) units to arrange in a row:

\[6!=720\]

Within the block, Edem and his wife can swap in \(2!=2\) ways. Total:

\[720\times 2=1440\ \text{ways}\]

(b) Bag of 4 red and 5 black (total 9), 5 draws with replacement, so \(P(R)=\dfrac49,\ P(B)=\dfrac59\).

(i) Red picked exactly 3 times:

\[{}^{5}C_3\left(\tfrac49\right)^{3}\left(\tfrac59\right)^{2}=10\cdot\frac{64}{729}\cdot\frac{25}{81}=\frac{16000}{59049}\approx 0.2710\]

(ii) Black picked at most 2 times (0, 1 or 2), with \(P(B)=\tfrac59\):

\[P(0)=\left(\tfrac49\right)^{5}=\frac{1024}{59049}\]\[P(1)={}^{5}C_1\left(\tfrac59\right)\left(\tfrac49\right)^{4}=\frac{6400}{59049}\]\[P(2)={}^{5}C_2\left(\tfrac59\right)^{2}\left(\tfrac49\right)^{3}=\frac{16000}{59049}\]\[P(\le 2)=\frac{1024+6400+16000}{59049}=\frac{23424}{59049}\approx 0.3967\]