From the 7 HODs of seven departments of an engineering college, one is to be promoted to be the principal of the college. Further, the position which is now vacant for the HOD is to be filled by an Associate Professor being promoted as an HOD. And further, one Assistant Professor is to be promoted to be an Associate Professor. No further promotions are made. The number of Associate Professors in the college is 4, 5, 3, 5, 2, 1 and 4 and that of Assistant Professors is 8, 9, 12, 5, 6, 7 and 6 for the departments of Mechanical, Electrical, Electronics, Civil, Chemical, Computer Science and Production Engineering, respectively. In how many different ways can the changes occur?a)1267b)181c)212d)None of the aboveCorrect answer is option 'B'. Can you explain this answer?

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From the 7 HODs of seven departments of an engineering college, one is to be promoted to be the principal of the college. Further, the position which is now vacant for the HOD is to be filled by an Associate Professor being promoted as an HOD. And further, one Assistant Professor is to be promoted to be an Associate Professor. No further promotions are made. The number of Associate Professors in the college is 4, 5, 3, 5, 2, 1 and 4 and that of Assistant Professors is 8, 9, 12, 5, 6, 7 and 6 for the departments of Mechanical, Electrical, Electronics, Civil, Chemical, Computer Science and Production Engineering, respectively. In how many different ways can the changes occur?

a)
1267
b)
181
c)
212
d)
None of the above

Correct answer is option 'B'. Can you explain this answer?

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From the 7 HODs of seven departments of an engineering college, one is...

To solve this problem, we need to find the number of different ways the changes can occur. Let's break down the problem step by step.

Step 1: Promoting a HOD to be the Principal
There are 7 HODs, and we need to choose 1 of them to be promoted as the Principal. This can be done in 7 ways.

Step 2: Filling the vacant HOD position
Once a HOD is promoted to be the Principal, there will be a vacant HOD position that needs to be filled by promoting an Associate Professor. There are 7 HODs to choose from, but one of them has already been promoted to be the Principal. So, we have 6 HODs left for this position. This can be done in 6 ways.

Step 3: Promoting an Assistant Professor to be an Associate Professor
After filling the vacant HOD position, there will be a vacant Associate Professor position that needs to be filled by promoting an Assistant Professor. We have 7 departments, each with a different number of Assistant Professors. We need to choose 1 department, and then choose 1 Assistant Professor from that department. This can be done in (7 * 1) = 7 ways.

Step 4: Calculating the total number of ways
To calculate the total number of ways, we multiply the number of ways from each step together: 7 * 6 * 7 = 294.

However, we need to consider that the number of Associate Professors in each department is different. We need to find the product of these numbers and multiply it with the previous result. The product of (4 * 5 * 3 * 5 * 2 * 1 * 4) = 480.

So, the total number of different ways the changes can occur is: 294 * 480 = 141,120.

This is not one of the options provided, so it seems there might be an error in the question or the options.

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