Five persons buy a thing one after another. Find how many persons buy the thing after T?

Statement I: One person buys the thing between R and Q. At least two persons buy the thing between S and R.

Statement II: Three persons buy the thing between Q and T. R buys exactly between Q and T. S does not buy just before or just after T. P buys the thing after S.

Question

Five persons buy a thing one after another. Find how many persons buy the thing after T?

Statement I: One person buys the thing between R and Q. At least two persons buy the thing between S and R.

Statement II: Three persons buy the thing between Q and T. R buys exactly between Q and T. S does not buy just before or just after T. P buys the thing after S.

Accepted Answer

**Solution:**

We need to find how many persons buy the thing after T.

Let's analyze the statements step-by-step.

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**Statement I:**
1. One person buys the thing between R and Q.
2. At least two persons buy the thing between S and R.

From this, we know the relative positions of R, Q, and S with some gaps, but not the exact order or the position of T or P. So, we cannot determine how many persons buy after T from this statement alone.

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**Statement II:**
1. Three persons buy the thing between Q and T.
2. R buys exactly between Q and T.
3. S does not buy just before or just after T.
4. P buys after S.

From this:
- Since three persons are between Q and T, the order is something like Q - x - x - x - T.
- R is exactly between Q and T, so R is the middle person among the three between Q and T.
- S is not adjacent to T.
- P buys after S.

However, we do not know the position of S relative to Q, R, T, or P exactly, nor the position of the fifth person.

Hence, statement II alone is insufficient.

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**Combining I and II:**
- From I, we know the relative positions and gaps between R, Q, and S.
- From II, we know the exact number of persons between Q and T and the position of R between them.
- Also, P buys after S.

Let's try to arrange:

Positions: 1, 2, 3, 4, 5 (order of buying)

From II:
- Q and T have 3 persons between them, so Q and T must be at positions 1 and 5 or 5 and 1.
- R is exactly in the middle between Q and T, so R is at position 3.

Assuming Q at position 1 and T at position 5:
- Positions: Q(1), ?, R(3), ?, T(5)

From I:
- One person between R and Q: Between positions 1 and 3, position 2 is the one person.
- At least two persons between S and R: So S must be at position 5 (since between S and R at 3, positions 4 and 5 are after R, but 5 is T, so S can't be 5), or position 1 (Q is there), or position 4.

Since T is at 5, S cannot be 5.

Try S at 4:
- Between S(4) and R(3) is only one person (none), so this contradicts 'at least two persons between S and R.'

Try S at 1 (Q is there), no.

Try S at 2:
- Between S(2) and R(3) is zero persons, no.

Try S at 5:
- T is at 5, so no.

Try Q at 5 and T at 1:
- Positions: T(1), ?, R(3), ?, Q(5)
- One person between R(3) and Q(5) is position 4.
- At least two persons between S and R(3): S must be at position 1 or 5.

S can't be 5 (Q is there), so S at 1 (T is there), no.

Try S at 2:
- Between S(2) and R(3) is zero persons, no.

Try S at 4:
- Between S(4) and R(3) is zero persons, no.

No valid arrangement with Q and T at ends.

Try T at 4 and Q at 1:
- Between Q(1) and T(4) are positions 2 and 3 (2 persons), but statement II says 3 persons between Q and T.

Try T at 5 and Q at 2:
- Between Q(2) and T(5) are positions 3 and 4 (2 persons), no.

Try T at 5 and Q at 1:
- Between Q(1) and T(5) are positions 2, 3, 4 (3 persons), good.
- R is exactly between Q and T, so R at 3.
- One person between R(3) and Q(1) is position 2.
- At least two persons between S and R(3): S must be at 5 or 4.

S cannot be 5 (T is there), so S at 4.
- P buys after S(4), so P at 5.

Order:
1: Q
2: ? (one person between R and Q)
3: R
4: S
5: P or T

T is at 5, so P must be at 5? No, T is at 5.

Conflict here.

Try P at 5, T at 4:
- But T is at 5 from statement II.

So T is at 5, P at 5 not possible.

Try P at 2:
- P buys after S(4), no.

Try P at 5:
- P buys after S(4), yes.

T is at 5, so P cannot be at 5.

Try P at 2:
- P after S(4), no.

Try P at 3:
- R is at 3.

Try P at 1:
- Q is at 1.

Try P at 4:
- S is at 4.

No place for P.

Hence, the only possible order is:
1: Q
2: P
3: R
4: S
5: T

Check conditions:
- One person between R(3) and Q(1) is P(2), correct.
- At least two persons between S(4) and R(3): only one person between 3 and 4, no.

Contradiction.

Try S at 5:
- T at 4, P after S(5) no.

Try S at 2:
- Between S(2) and R(3) zero persons, no.

Try S at 1:
- Q at 1, no.

No arrangement satisfies all conditions.

Therefore, both statements together are necessary to answer the question, but even then, the exact number of persons after T cannot be determined with certainty.

Hence, the answer is (e): Both statements together are necessary to answer.

**Answer:** (e) If the data in both the statements I and II together are necessary to answer

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