A rule like these two cannot sit together changes the count. Two reliable moves: place the restricted objects first and fill in around them, or count everything and subtract the arrangements that break the rule.
●●●5 points, difficulty 3 of 35 ptLevel 5–6about 135s
Five children sit in a row. Two of them are twins who insist on sitting next to each other. In how many orders can the five sit?
Hints
Take them one at a time. The first gives nothing away.
1A nudge
Glue the twins together into one block.
2The strategy
Treat the pair as a single object, so you are arranging four things rather than five. Then remember the two inside the block can be in either order.
3The full solution
Four objects arrange in 4 x 3 x 2 x 1 = 24 ways, and the twins can be either way round inside the block, so 24 x 2 = 48.
Solution
The reliable way
Tie the twins together and treat them as one object. You now have four objects to arrange, which can be done in 4 x 3 x 2 x 1 = 24 ways. Inside the block the twins can sit in 2 orders, so the total is 24 x 2 = 48. Sanity check: 48 is less than the 120 unrestricted arrangements, as it must be. The transferable idea: place the restricted objects first by gluing them, then multiply by their internal orders.
The elegant way
The twins occupy one of 4 adjacent pairs of seats, in 2 orders, and the other three children fill the rest in 3 x 2 x 1 = 6 ways: 4 x 2 x 6 = 48.
Why this is on the test: The glue-them-together move is the standard technique for a must-sit-together restriction, and forgetting the internal swap halves the answer.
Where it goes wrong
Each of these is written the way a student would say it to themselves.
“I counted every arrangement and forgot to remove the ones that break the rule.”
“I placed the free objects first and then found the restricted ones would not fit.”
“I treated the pair that must stay together as two separate objects.”
Every problem in this skill
●○○3 points, difficulty 1 of 33 ptOne step from a skill you know.
Nothing at this tier yet for CNT-11.
●●○4 points, difficulty 2 of 34 ptTwo steps, or one step plus a trap.
Nothing at this tier yet for CNT-11.
●●●5 points, difficulty 3 of 35 ptNeeds an idea, not just a procedure.