9 persons enter a lift from ground floor of a building which stops in 10 floors (excluding ground floor), if it is known that persons will leave the lift in groups of 2,3, & 4 in different floors. In how many ways this can happen?

Answer

To solve the problem of how many ways 9 persons can exit a lift in groups of 2, 3, and 4 on different floors, we can break down the solution into several steps.

### Step 1: Determine the Groups
We have 9 persons who will exit in groups of 2, 3, and 4. This means we need to identify how many ways we can choose these groups from the 9 persons.

### Step 2: Choose the Group of 2
We first choose 2 persons out of 9 to form the first group. The number of ways to choose 2 persons from 9 is given by the combination formula:

\[ \text{Number of ways to choose 2 persons} = \binom{9}{2} = \frac{9!}{2!(9-2)!} = \frac{9 \times 8}{2 \times 1} = 36 \]

### Step 3: Choose the Group of 3
Next, we need to choose 3 persons from the remaining 7 persons (after choosing the first group). The number of ways to choose 3 persons from 7 is:

\[ \text{Number of ways to choose 3 persons} = \binom{7}{3} = \frac{7!}{3!(7-3)!} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35 \]

### Step 4: Choose the Group of 4
Finally, the remaining 4 persons will automatically form the last group. The number of ways to choose 4 persons from 4 is:

\[ \text{Number of ways to choose 4 persons} = \binom{4}{4} = 1 \]

### Step 5: Calculate the Total Combinations of Groups
Now, we multiply the number of ways to choose each group:

\[ \text{Total ways to form groups} = \binom{9}{2} \times \binom{7}{3} \times \binom{4}{4} = 36 \times 35 \times 1 = 1260 \]

### Step 6: Arrange the Groups
Next, we need to consider the arrangement of these groups. Since we have 3 groups (of sizes 2, 3, and 4), they can be arranged in:

\[ \text{Number of arrangements of groups} = 3! = 6 \]

### Step 7: Calculate the Total Ways
Finally, we combine the number of ways to form the groups and the arrangements of these groups:

\[ \text{Total ways} = 1260 \times 6 = 7560 \]

### Conclusion
Thus, the total number of ways the 9 persons can exit the lift in groups of 2, 3, and 4 on different floors is **7560**.

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