Q80 / 180 · COMEDK 2023 (Evening Shift) · Mathematics
In a 12 storey house, 10 people enter a lift cabin. It is known that they will leave the lift in pre-decided groups of 2, 3 & 5 people at different storeys. The number of ways they can do so if the lift does not stop up to the second storey is
- A78
- Correct answerB720
- C132
- D120
Solution
The total number of storeys available is 12. Since the lift does not stop at the first and second storeys, the available storeys for the groups to exit are from the 3rd to the 12th storey. This gives $12 - 2 = 10$ available storeys.
There are 3 groups of people consisting of 2, 3, and 5 individuals. These groups must exit at different storeys. We need to select 3 distinct storeys out of the 10 available storeys and assign the groups to them.
The number of ways to choose 3 storeys out of 10 is $^{10}C_{3} = \dfrac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120$.
Once the 3 storeys are chosen, the 3 distinct groups (size 2, 3, and 5) can be arranged in these 3 storeys in $3!$ ways.
Total number of ways = $^{10}C_{3} \times 3! = 120 \times 6 = 720$.
Answer: 720
There are 3 groups of people consisting of 2, 3, and 5 individuals. These groups must exit at different storeys. We need to select 3 distinct storeys out of the 10 available storeys and assign the groups to them.
The number of ways to choose 3 storeys out of 10 is $^{10}C_{3} = \dfrac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120$.
Once the 3 storeys are chosen, the 3 distinct groups (size 2, 3, and 5) can be arranged in these 3 storeys in $3!$ ways.
Total number of ways = $^{10}C_{3} \times 3! = 120 \times 6 = 720$.
Answer: 720
Hence, the correct answer is option: 'B'.