How many different four​-letter passwords can be formed from the letters Upper A comma Upper B comma Upper C comma Upper D comma Upper E comma Upper F comma and Upper G if no repetition of letters is​ allowed?

Accepted Solution

Answer with explanation:Number of Letters from which we have to make four​-letter passwords    ={A,B,C,D,E,F,G}=7Since repetition of digit is not allowed.In Permutation order of arrangement is Important,while in combination order of arrangement is not Important.Out of seven letters we have to select four letters, keeping in mind the order of Alphabets.So, we will use the concept of Permutation.Number of ways of arrangement,that is to make four letter password from 7 Alphabets        [tex]_{4}^{7}\textrm{P}\\\\=\frac{7!}{(7-4)!}\\\\=\frac{7!}{3!}\\\\=4\times 5\times 6 \times 7=840[/tex]So, total number of ways of making password from 7 alphabets=840 waysSecond Method⇒Since repetition of alphabets is not allowed, first Alphabet can be chosen in 7 ways , second alphabet can be chosen in 6 ways, third alphabet can be chosen in 5 ways, fourth alphabet can be chosen from four alphabets.So, total number of ways of making four letter password        =7×6×5×4     = 840 ways