Question

There are eight different jobs in a printer queue. Each job has a distinct tag which is a string of three upper case letters.

The tags for the eight jobs are: { LPW, QKJ, CDP, USU, BBD, PST, LSA, RHR }

How many different ways are there to order the eight jobs in the queue so that job USU comes somewhere before CDP in the queue (although not necessarily immediately before) and CDP comes somewhere before BBD (again, not necessarily immediately before)?

Answer 1

Job USU comes somewhere before CDP in the queue, although not necessarily immediately before CDP.

Case 1:

If CDP is in the 1^st position, there is no chance to job USU to be in the before position CDP.

Case 2:

If CDP is in the 2^nd position, then job USU must be in position 1 to be in the before position CDP. Then, the remaining 6 jobs can be ordered in 6! Ways.

Case 3:

If CDP is in the 3^rd position, then job USU must be in position 1 or position 2 to be in the before position CDP.

If USU is position 1, then the remaining 6 jobs can be ordered in 6! ways.

If USU is position 2, then the remaining 6 jobs can be ordered in 6! ways.

Then, the total ways = 2x 6! Ways.

Case 4:

If CDP is in the 4^th position, then job USU must be in position 1, position 2, or position 3 to be in the before position CDP.

If USU is position 1, then the remaining 6 jobs can be ordered in 6! ways.

If USU is position 2, then the remaining 6 jobs can be ordered in 6! ways.

If USU is position 3, then the remaining 6 jobs can be ordered in 6! ways.

Then, the total ways = 3x 6! Ways.

Case 5:

If CDP is in the 5^th position, then job USU must be in position 1, position 2, position 3, or position 4 to be in the before position CDP.

Then, the total ways = 4x 6! Ways.

Case 6:

If CDP is in the 6^th position, then job USU must be in positions 1,2,3,4, or 5 to be in the before position CDP.

Then, the total ways = 5x 6! Ways.

Case 7:

If CDP is in the 7^th position, then job USU must be in positions 1,2,3,4,5 or 6 to be in the before position CDP.

Then, the total ways = 6x 6! Ways.

Case 8:

If CDP is in the 8^th position, then job USU must be in positions 1,2,3,4,5,6 or 7 to be in the before position CDP.

Then, the total ways = 7x 6! Ways.

Therefore, the total number of ways is 6! +(2x 6!) +(3x 6!) +(4x 6!) +(5x 6!) +(6x 6!) +(7x 6!) = (1+2+3+4+5+6+7) x 6! = 28 x 6!

Hence, the number of different ways is there to order the eight jobs in the queue so that job USU comes somewhere before CDP is 28 x 6! = 20160 Ways.

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