There are 5 doors to a lecture room . The number of ways that a student can enter the room and leave it by a different door is ? (a) 20 (b) 16 (c) 19 (d) 25?

Question

Answer

Answer

While entering the student can choose any one of the 5 doors and while exiting he can choose any one of the remaining 4 doors. So, total ways = 5 × 4 = 20 ways

Let's try some other way.
The student can enter the room in 5 ways and can exit it in 5 ways. So, the number of ways in which he can enter aand exit the room in 5 × 5 = 25 ways.
Now out of the 25 ways in some of them the student may have enetered and exited through the same door. He can enter and exit through the same door from any one of the 5 doors. So there will be 5 ways in which he enter an exit through the same door.
Total ways he can enter and exit through a different gate = 25 - 5 = 20

Answer

**Solution:**

To find the number of ways a student can enter the room and leave it by a different door, we can break down the problem into two steps:

Step 1: Entering the room
Step 2: Leaving the room

Let's consider each step in detail:

**Step 1: Entering the Room**

Since there are 5 doors to the lecture room, the student has 5 options to choose from while entering the room.

**Step 2: Leaving the Room**

After entering the room, the student needs to leave through a different door. Since the student cannot leave through the same door they entered, they have 4 remaining doors to choose from.

Therefore, the number of ways the student can enter the room and leave it by a different door is the product of the number of options in each step:

Number of ways = Number of options for entering the room × Number of options for leaving the room

Number of ways = 5 (options for entering) × 4 (options for leaving)

Number of ways = 20

Hence, the correct answer is (a) 20.

In summary:

- The student has 5 options to choose from while entering the room.
- After entering, the student has 4 remaining doors to choose from while leaving.
- The number of ways the student can enter and leave the room by a different door is 5 × 4 = 20.

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