**Step 1** |

The first step is finding out all the possible arrangements of the letters of the word MATHEMATICS. Note that we are not interested in "unique" arrangements (in some kind of problems we would be), but just the total possible arrangement |

**Step 2** |

The answer to this is that there are 11 letters in this word, and therefore the letters can be arranged in **11!** ways i.e. 11x10x9x...2x1 ways
To see this, take any of the letters - it can be in any of the possible 11 positions.
For each of these position, a second letter can be in any of the other 10 positions (11 minus the one taken up by the first letter)
So the two letters can appear in 11x10 combinations
For each of these 110 combinations, a third letter can be in any of the remaining 9 places, and so on |

**Step 3** |

Now we know the total number of arrangements (**11!**) possible, and need to look for the possible arrangements where the letters H and S have exactly 4 letters between them) |

**Step 4** |

This can be seen by inspection
If H is in the first position (first letter of the word), then S would need to be in position 6 (since there are 4 letters between them
Similarly, if H is in the second position , then S would need to be in position 7
There are 6 such positions, the last one having H in the 6th position and S being in the last position |

**Step 5** |

The same thing can be seen with S being before H. There are 6 such positions
So the total number of positions for the two letters where this condition is met is 12
Now we have filled in 2 of the 11 letters with H and S in 12 ways
The remaining 9 letters can take any of the remaining 9 positions for each of these
Since there are no restriction on the remaining 9 letters, the number of possible arrangements of 9 letters in 9 positions is **9!**
So the total ways to rearrange all the letters so that H and S have exactly 4 letters between them is
**12 x 9! ** |

**Step 6** |

Now we can work out the probability of rearranging the letters of MATHEMATICS so that H and S have exactly 4 letters between them
P (arrangement) = Arrangements where the two letters have 4 letters between them | Total possible arrangements of the letters of MATHEMATICS | = = |