KA-0070Counting with restrictions
5 points, difficulty 3 of 3Level 5–6about 135sFive 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.