Directions (81-85): Study the following information carefully and answer the questions given below:

Eight persons buy different things one after other. Two persons buy between M and L who buys immediately before Q. O buys two persons before Q. As many persons buy between O and Q as between L and K. K buys before O. Two persons buy between K and P. J buys before N and after P.

If all the persons buy things in the alphabetical order from bottom to top then the position of how many persons remains unchanged?

To find the correct ordering of the eight persons (J, K, L, M, N, O, P, Q) buying things, let us analyze the clues step-by-step to determine their positions from 1 (first to buy, top) to 8 (last to buy, bottom):

1. "L buys immediately before Q":
This means L and Q are consecutive, represented as the block: [L, Q].

2. "O buys two persons before Q":
Buying "two persons before" means there is exactly one person between O and Q. Since L is immediately before Q, O must be immediately before L. Thus, we have the block: [O, L, Q].

3. "As many persons buy between O and Q as between L and K":
From our block [O, L, Q], there is exactly 1 person (L) between O and Q. Therefore, there must be exactly 1 person between L and K. This gives two possibilities for K's position relative to L:
- Case A: K is two positions before L (i.e., [K, _ , L]). Since O is immediately before L, this means K is immediately before O, giving the block [K, O, L, Q].
- Case B: K is two positions after L (i.e., [L, _ , K]). Since Q is immediately after L, this means K is immediately after Q, giving the block [O, L, Q, K].

4. "K buys before O":
Comparing the two cases:
- In Case A [K, O, L, Q], K buys before O. This is consistent with the clue.
- In Case B [O, L, Q, K], K buys after O. This violates the clue. Hence, Case B is eliminated.
So, we have the established block of four consecutive positions: [K, O, L, Q].

5. "Two persons buy between K and P":
Using our block [K, O, L, Q], let us look at the placement of P relative to K:
- If P is before K, then the sequence would be [P, _ , _ , K, O, L, Q]. Since K is followed by 3 persons, this sequence would contain 3 (before K) + 4 (K to Q) = 7 persons. Let's list their positions: 1: P, 2: [empty], 3: [empty], 4: K, 5: O, 6: L, 7: Q, 8: [empty].
- If P is after K, then there are 2 persons between K and P. In our block [K, O, L, Q], the two persons after K are O and L. Therefore, P must be the next person, which is immediately after L. But Q is already immediately after L. This is a contradiction. Thus, P must buy before K.

This gives the configuration: [P, _ , _ , K, O, L, Q]. Since there are 8 persons in total, the remaining position can be either at the very beginning (before P) or at the very end (after Q). Let's test these two possibilities:

Possibility 1: The remaining position is after Q (positions 1 to 8):
1: P
2: [empty]
3: [empty]
4: K
5: O
6: L
7: Q
8: [empty]

Possibility 2: The remaining position is before P (positions 1 to 8):
1: [empty]
2: P
3: [empty]
4: [empty]
5: K
6: O
7: L
8: Q

6. "Two persons buy between M and L":
Let's check this constraint against both possibilities:
- In Possibility 1, L is at position 6. The positions two persons away from L are 3 (two positions before) and 9 (two positions after, which is out of bounds). Therefore, M must be at position 3. This is valid.
- In Possibility 2, L is at position 7. The positions two persons away from L are 4 (two positions before) and 10 (out of bounds). Therefore, M must be at position 4. This is also valid.

7. "J buys before N and after P":
Let's apply this to the remaining empty slots for each possibility:
- For Possibility 1 (M at 3):
1: P
2: [empty]
3: M
4: K
5: O
6: L
7: Q
8: [empty]
The empty slots are 2 and 8. Since J must buy before N, J must be at position 2 and N must be at position 8. Also, J (at 2) is indeed after P (at 1). This is a completely valid arrangement:

Final Buying Order (from Top to Bottom):
1: P
2: J
3: M
4: K
5: O
6: L
7: Q
8: N

Let's double-check Possibility 2 to see if it can also be valid:
- For Possibility 2 (M at 4):
1: [empty]
2: P
3: [empty]
4: M
5: K
6: O
7: L
8: Q
The empty slots are 1 and 3. Since J must buy after P (who is at 2), J cannot be at position 1. Therefore, J must be at position 3. Consequently, N must be at position 1. But this violates the condition that J buys before N. Hence, Possibility 2 is invalid.

Now, let's compare the actual order with the alphabetical order arranged from bottom to top (Position 8 to Position 1):

Alphabetical Order from Bottom to Top:
The alphabetical order of the eight persons is J, K, L, M, N, O, P, Q.
Arranging them from bottom (position 8) to top (position 1):
1: Q
2: P
3: O
4: N
5: M
6: L
7: K
8: J

Let's compare the two tables side-by-side to find whose positions remain unchanged:
- Position 1: P vs Q (changed)
- Position 2: J vs P (changed)
- Position 3: M vs O (changed)
- Position 4: K vs N (changed)
- Position 5: O vs M (changed)
- Position 6: L vs L (Unchanged)
- Position 7: Q vs K (changed)
- Position 8: N vs J (changed)

Only the position of L (at position 6) remains unchanged. Therefore, the position of One person remains unchanged.