Video Transcript
How many four-digit numbers can be
formed from the digits five, three, two, seven, and six? Assume no number can be used more
than once.
In this question, we need to select
four-digit numbers from the five digits five, three, two, seven, and six. For example, we could pick the
number 5327. We are told that the digits cannot
be used more than once. However, order does matter, as the
number 5372 is different to the number 5327.
Trying to list all these numbers
would be very time consuming. So instead we will use our
knowledge of permutations. A permutation, denoted 𝑛𝑃𝑘,
represents the number of different ways to order 𝑘 objects from 𝑛 total distinct
objects. And when dealing with permutations,
the order of the elements matter. In this question, since there are
five digits altogether and we’re selecting four of them, we need to calculate five
𝑃 four.
We know that 𝑛𝑃𝑘 is equal to 𝑛
factorial divided by 𝑛 minus 𝑘 factorial. This means that five 𝑃 four is
equal to five factorial divided by one factorial. Five factorial is equal to five
multiplied by four multiplied by three multiplied by two multiplied by one. And one factorial is simply equal
to one. Five 𝑃 four is therefore equal to
120. There are 120 different four-digit
numbers that can be formed from the five digits five, three, two, seven, and
six. This answer would be true whatever
the five numbers as long as they were unique.
