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There are 5 doors to a lecture room. Two...

There are 5 doors to a lecture room. Two are red and the others are green. In how many ways can a lecturer enter the room and leave the room from different colored doors?

A

1

B

3

C

6

D

12

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem of how many ways a lecturer can enter and leave a lecture room through different colored doors, we can break it down into steps. ### Step 1: Identify the doors There are a total of 5 doors: - 2 Red doors (let's call them R1 and R2) - 3 Green doors (let's call them G1, G2, and G3) ### Step 2: Consider entering through a red door If the lecturer enters through one of the red doors, he has 2 options: - Enter through R1 - Enter through R2 #### Case 1: Entering through R1 - If he enters through R1, he can exit through any of the 3 green doors (G1, G2, G3). This gives us 3 options. #### Case 2: Entering through R2 - If he enters through R2, he can again exit through any of the 3 green doors (G1, G2, G3). This also gives us 3 options. So, if he enters through a red door, the total number of ways to enter and exit is: - Total for red doors = 3 (from R1) + 3 (from R2) = 6 ways. ### Step 3: Consider entering through a green door If the lecturer enters through one of the green doors, he has 3 options: - Enter through G1 - Enter through G2 - Enter through G3 #### Case 3: Entering through G1 - If he enters through G1, he can exit through either of the 2 red doors (R1, R2). This gives us 2 options. #### Case 4: Entering through G2 - If he enters through G2, he can exit through either of the 2 red doors (R1, R2). This also gives us 2 options. #### Case 5: Entering through G3 - If he enters through G3, he can exit through either of the 2 red doors (R1, R2). This again gives us 2 options. So, if he enters through a green door, the total number of ways to enter and exit is: - Total for green doors = 2 (from G1) + 2 (from G2) + 2 (from G3) = 6 ways. ### Step 4: Combine the totals Now, we can combine the totals from both cases: - Total ways = Ways from red doors + Ways from green doors = 6 + 6 = 12 ways. ### Final Answer Thus, the total number of ways the lecturer can enter and leave the room through different colored doors is **12**. ---
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