# Practice Paper 3 Question 20

The numbers $$1, 2, ..., n$$ are permuted (or shuffled). How many different permutations exist such that no two of the numbers $$1, 2, 3$$ are adjacent when $$n = 5$$ and $$n = 6$$? How about for arbitrary $$n > 4$$?

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## Warm-up Questions

1. How many distinct orderings of the letters in the string "AAABBB" are there?
2. In how many ways can you order the sequence of digits $$1,2,3,4,5$$ such that $$1$$ always comes before $$2?$$

## Hints

• Hint 1
How many ways are there to shuffle numbers $$4,5, \ldots, n?$$
• Hint 2
How many ways are there to shuffle numbers $$1,2,3?$$
• Hint 3
How many ways are there to pick three non-adjacent points to insert $$1,2,3$$ into the other numbers?

## Solution

We start by taking the numbers $$4,5, \ldots, n$$ and shuffling them. There are $$(n-3)!$$ ways of doing this. There are $$n-2$$ slots that are separated by the $$n-3$$ shuffled numbers, and if we insert each of $$1,2,3$$ into a different slot, they cannot be adjacent. There are $$\binom{n-2}{3}$$ ways to do this. Finally, there are $$3!$$ ways to order the numbers $$1,2,3$$. Multiplying these together we get: \begin{align} 6 \, \binom{n-2}{3} \, (n-3)! &= \frac{6\, (n-3)! \, (n-2)!}{6\,(n-5)!} \\ &= (n-3)(n-4)(n-2)!. \end{align}

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