Question

The letters “A B C D E F G” are used to form a five letter secret code. Repetitions are NOT allowed. How many different codes are possible?

Answer 1

Given letters are A, B, C, D, E, F and G.

There are 7 letters.

n = 7

We have to form different '5 letter' secret codes.

r = 5

Given that repetition is not allowed.

Thus the total no. of possible codes is,

$\displaystyle \Large \text{ \mathsf{\ ^nP_r =\ ^7P_5 \ =\ \frac{7!}{(7-5)!}\ = \ \frac{7!}{2!}\ =\ \frac{5040}{2}\ =\ \bold{2520}} }$

⇒  We have to form five letter codes. The given 7 letters can be used at 1st place.

⇒  As letter repetition is not occurred, if one letter is used in 1st place, the other 6 letters can be used at 2nd place.

⇒  If two letters are used at 1st and 2nd places, the other 5 letters can be used at 3rd place.

⇒  4 letters can be used at 4th place.

⇒ 3 letters can be used at 5th place.

Hence the total no. possible codes is 7 × 6 × 5 × 4 × 3 = 2520.