#### In how many ways can the letters of the word "ENGINEER" be arranged such that the N and E should not be together?Option: 1 2,127,253Option: 2 9,916,656Option: 3 2,902,040Option: 4 8,258,963

To calculate the number of ways the letters of the word "ENGINEER" can be arranged such that the letters N and E are not together, we can treat the N and E as distinct entities and count the number of valid arrangements.

The word "ENGINEER" has a total of 8 letters, but since the N and E are treated as distinct entities, we have a total of 8 + 2 = 10 objects to arrange.

We can arrange these 10 objects in 10! ways. However, within this arrangement, we have to account for the cases where the N and E are together.

Consider the NE as a single entity. We can arrange these 9 entities ("NE, G, I, N, E, E, R") in 9! ways. Within each of these arrangements, the N and E can be rearranged among themselves in 2! ways.

Therefore, the total number of arrangements where the N and E are not together is given by:

$10 !-(9 ! \times 2 !)=3,628,800-(362,880 \times 2)=3,628,800-725,760=2,902,040$

Thus, there are 2,902,040 ways to arrange the letters of the word "ENGINEER" such that the letters N and E are not together.