Question

Outside a home, there is a 10- key pad with letters A, B,C,D,E,F,G,H,I and J that can be used to open a garage door if the correct ten letter code s entered each key can only be used once. How many codes are there?

Answer 1

Answer:

There are 3,628,800 possible codes.

Step-by-step explanation:

Observe that:

1. For the first letter in the code there are 10 possible choices.
2. For the second letter in the code we can not use the letter used in 1). Then, we have only 9 possible choices.
3. For the third letter in the code we can not use the letter used in 1) neither the one used in 2). Then we have only 8 possible choices.

Following this argument we can see that the number of possible choices for the 4th,5th,6th,7th,8th,9th and 10th letter are 7,6,5,4,3,2 and 1 respectively.

Finally, since the letters choice are independent, we have that the number of possible codes is given by

$10!=10\times9\times...\times1=3,628,800.$