Raman and Sanjay are members of a quiz team which has 9 members in total. Two photographs of the quiz team are to be taken.
For the second photograph, the members will stand in two rows, with 5 in the back row and 4 in the front row.
(c) In how many different ways can the 9 members be divided into a group of 5 and a group of 4?
(d) For a random division into a group of 5 and a group of 4, find the probability that Raman and Sanjay are in the same group as each other.
Solution
(c) To divide 9 members into a group of 5 and a group of 4, we choose 5 members out of 9. This can be calculated using combinations:
\(\binom{9}{5} = 126\)
(d) To find the probability that Raman and Sanjay are in the same group, consider two cases:
1. Both are in the group of 5. Choose 3 more members from the remaining 7:
\(\binom{7}{3} = 35\)
2. Both are in the group of 4. Choose 2 more members from the remaining 7:
\(\binom{7}{2} = 21\)
Total ways for both to be in the same group:
\(35 + 21 = 56\)
Probability that Raman and Sanjay are in the same group:
\(\frac{56}{126} = \frac{4}{9} \approx 0.444\)
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