# If Yoshio is not assigned to the project, which one of the following could be true?

dionnamarcue on May 14, 2020

Help!

Victoria on September 16, 2020

Hi @dionnamarcue,

Happy to help!

We know that there are six students available for the history project: L, M, O, R, T, and Y.

Only four students will be assigned, and they will search archives from the years 1921, 1922, 1923, and 1924 (or, for the sake of ease, 1, 2, 3, and 4) with exactly one student searching each archive.

_ _ _ _ | _ _
1 2 3 4

Now let's go through our conditions.

Rule 1 - only L or T can be assigned to 3

3 --> L or T
Not L and Not T --> Not 3

_ _ L/T _ | _ _
1 2 3 4

Rule 2 - if M is assigned to the project, then she must be assigned to either 1 or 2

M --> 1 or 2
Not 1 and Not 2 --> Not M

Rule 3 - if T is assigned to the project, then R must be assigned to the project

T --> R
Not R --> Not T

Rule 4 - if R is assigned to the project, then O must be assigned to the year immediately prior to R's

R --> |OR|
Not |OR| --> Not R

Notice that we can combine Rules 3 and 4 to make a transitive chain.

T --> R --> |OR|
Not |OR| --> Not R --> Not T

This chain and Rule 1 together are pretty restrictive. So, let's try to map out some possible scenarios before we address the question stem.

Rule 1 tells us that only L or T can be assigned to 3. Therefore, we can split up our scenarios based on who is assigned to 3.

There is only one main possibility if T is assigned to 3 because of the restrictive transitive chain. However, there are a couple scenarios if L is assigned to 3 because Rules 3 and 4 impose conditions if T and R are assigned, but do not require them to be assigned.

Option 1: T is assigned to 3

We know that if T is assigned, then R must also be assigned, and O is assigned to the year immediately prior to R's. Therefore, R must be assigned to 2 and O must be assigned to 1.

We also know that, if M is assigned, then she must be assigned to either 1 or 2. Both of these spaces are filled by O and R; therefore, M cannot be assigned.

O R T L/Y | M Y/L
1 2 3 4

Option 2 - L is assigned to 3 and R is assigned to the project.

Again, R must be assigned to 2 and O must be assigned to 1, so M cannot be assigned.

O R L T/Y | M Y/T
1 2 3 4

|OR| is extremely restrictive, so our final scenario will eliminate this restriction.

Option 3 - L is assigned to 3 and O is not assigned immediately prior to R

Therefore, R is not assigned to the project. This means that T cannot be assigned to the project either, but it also means that M can be assigned to the project.

M/O/Y O/Y/M L Y/O | T R
1 2 3 4

There are no other possible scenarios because the inclusion of |OR| when L is assigned to 3 restricts us to Option 2 whereas their exclusion restricts us to Option 3. Therefore, Options 1 through 3 as outlined above are the only possible scenarios.

Now let's address the question stem.

The question stem imposes an additional condition: Y is not assigned to the project. This narrows our choices down to L, M, O, R, and T. Four of these five students must be assigned to the project.

We can see that Options 1 and 2 above are possible if Y is not assigned whereas Option 3 is not. So, we are only focused on the first two.

Answer choice (A) is incorrect because Option 2 requires the inclusion of L and Option 1 requires the inclusion of L if Y is not assigned.

Answer choice (B) is incorrect because we have eliminated Option 3 and both Options 1 and 2 require that R is assigned.

Answer choice (C) is incorrect because Option 1 requires the inclusion of T and Option 2 requires the inclusion of T if Y is not assigned.

Answer choice (D) is incorrect because Rule 4 requires O to be assigned immediately before R if R is assigned. We know from eliminating answer choice (B) that R must be assigned. O cannot be assigned to 2 because 3 is filled by either T or L. Therefore, R must be assigned to 2 and O must be assigned to 1.

Answer choice (E) is correct because it is reflected in Option 1. If Y cannot be assigned, then this option requires:

O R T L | M Y
1 2 3 4

Therefore, L can be assigned to 4, making this our correct answer.

Hope this helps! I recognize this is a bit long and convoluted, so please let me know if you need any further clarification.