A school teacher must schedule seven sessions, which are abbreviated M, N, O, P, S, T, and U, during a day. Seven different consecutive time periods are available for the sessions, and are numbered one through seven in the order that they occur. Only one session can be schedules for each period. The assignment of the sessions to the periods is subject to the following restrictions: M and O must occupy consecutive periods. M must be scheduled for an earlier period than U.
O must be scheduled for a later period than S.
If S does not occupy the fourth period, then P must occupy the fourth period.
U and T cannot occupy consecutively numbered periods.

Which of the following could be true?
a. M is assigned to the first period.
b. O is assigned to the fifth period.
c. S is assigned to the seventh period.
d. T is assigned to the sixth period.

Given:

There are seven sessions to be scheduled in seven different consecutive time periods. Only one session can be scheduled for each period. The sessions are abbreviated as M, N, O, P, S, T, U. The restrictions are:

(i) M & O must occupy consecutive periods.

(ii) M must be scheduled for an earlier period than U.

(iii) O must be scheduled for a later period than S.

(iv) If S does not occupy the fourth period, then P must occupy the fourth period.

(v) U & T cannot occupy consecutively numbered periods.

Solution:

We will construct the sequence of sessions based on the given restrictions.

Since M & O must occupy consecutive periods, we can have the sequence as {..., M, O, ...} or {..., O, M, ...}

Since M must be scheduled for an earlier period than U, we can have the sequence as {..., M, O, ..., U, ...} or {..., O, M, ..., U, ...}

Since O must be scheduled for a later period than S, we can have the sequence as {..., S, ..., M, O, ..., U, ...} or {..., S, ..., O, M, ..., U, ...}

We can see that, according to the given restrictions, M cannot be assigned the first period as S has to be assigned before M. Thus option (a) is incorrect.

We can see that, according to the given restrictions, S cannot be assigned to the seventh period as seventh period is the last period and M, O & U has to be assigned after S. Thus option (c) is incorrect.

Now, T can be assigned in the following ways:

(I) After U: In this case, there are at least 4 sessions before T, the last of which is U. Moreover, according to the given restrictions, U & T cannot occupy consecutive periods. Also, since we are assuming that S is the first element, the fourth element has to be P, so that U is assigned to 5th period or after. Thus T has to be assigned to 7th, if we skip the period after U. That is, T cannot be assigned to the 6th period in this case.

(II) Between O, M & U: Even if U is assigned to the last (7th) period, since U & T cannot occupy consecutive periods, T cannot be assigned the 6th period in this case.

(III) Between S & O, M: This would imply that there are at least 3 sessions after T. This would automatically imply that T cannot be assigned to the 6th period in this case.

(IV) Before S: This implies that there are at least 4 sessions after T. Thus, T cannot be assigned to the 6th period in this case either.

Thus, T cannot be assigned to the sixth period in any case. That is, option (d) is incorrect.

Now, following all the given restrictions, one of the arrangements can be,

{1-N, 2-P, 3-T, 4-S, 5-O, 6-M, 7-U}

We can see that S is occupying the 4th period & U and T are not occupying consecutive periods. Thus, all the restrictions are followed. We can see that it is possible for O to be assigned to fifth period by following all the restrictions. Thus option (b) is the correct choice.

Final

Option (b) is the correct choice. That is, based on the given restrictions, O can be assigned to the fifth period.

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explanation: