# An almighty coincidence

Issue 45Imagine picking the four hymn numbers out of a hat. First note that four-hymn combinations with one 1-digit number and three 3-digit numbers come in four types: the one digit number can occur in first, second, third or last place of the selection. So the overall chance of picking such a combination is equal to:

Chance of picking a combination with 1-digit number in 1st place |

+ chance of picking a combination with 1-digit number in 2nd place |

+ chance of picking a combination with 1-digit number in 3rd place |

+ chance of picking a combination with 1-digit number in 4th place. |

Each of the terms in this sum is equal to

Now for the chance of picking a combination with two 2-digit numbers and two 3-digit numbers. There are 6 ways in which to choose the positions of the two 2-digit numbers within the string of four numbers, so this time the selection comes in 6 different types:

The chance of picking a combination of each individual type is

In general, the number of ways you can choose a set of positions within a sequence of length is