How many distinct four-digit numbers contain the digits 1, 2, 3, and 4 without repetition, if the digit 2 is always immediately followed by the digit 3?

1 Answer
Sep 2, 2016

#3! = 6#

Explanation:

Instead of individual digits, think of strings:

#a = #"#1#"

#b = #"#23#"

#c = #"#4#"

There are #3! = 3xx2xx1 = 6# ways to arrange the letters #a#, #b# and #c#.

Each such arrangement corresponds to a string of #4# digits satisfying the conditions of the question when concatenated.

For example:

#bac <=> #"#2314#"

Conversely, if the conditions of the question are satisfied, then a valid sequence of #4# digits can be represented by the letters #a#, #b# and #c# in some order.