Question:hard

There are eight persons, P, Q, R, S, T, U, V and W, standing in a row and four distinct articles A, B, C and D are to be given to four people. No four neighbouring persons receive an article. How many ways can this distribution be done?

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Count the ways to choose 4 positions out of 8 in a row, remove the cases where all four chosen positions are consecutive, then arrange the four articles.
Updated On: Jul 21, 2026
  • 1680 ways
  • 1560 ways
  • 1440 ways
  • 1380 ways
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The Correct Option is B

Solution and Explanation

Step 1: Count all ways to assign the 4 distinct articles without any restriction.
Handing 4 distinct articles to 4 different people chosen out of 8, one each, is a permutation of 8 taken 4 at a time.
$P(8,4) = 8\times7\times6\times5 = 1680$.

Step 2: Count the arrangements that break the rule.
The rule is broken only when the 4 people who get an article are four neighbours in a row.
There are 5 such neighbouring blocks in a row of 8: positions 1 to 4, 2 to 5, 3 to 6, 4 to 7, and 5 to 8.
For each block, the 4 distinct articles can be handed out in $4! = 24$ ways.
So the broken cases total $5 \times 24 = 120$.

Step 3: Remove the broken cases from the total.
Valid distributions $= 1680 - 120 = 1560$.

Final Answer:
The four articles can be distributed in 1560 valid ways. \[ \boxed{1560} \]
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