Question

Each of the letters A, H, I, M, O, T, U, V, W, X, Y appears same when looked at in the mirror. (No lateral inversion). They are hence called symmetric letters. Other letters in the alphabet are asymmetric letters. How many 4 letter computer passwords can be formed (no repetition allowed) with at least one symmetric letter?

• A. 11770
• B. 12540
• C. 326040
• D. 13585

Answer 1

The correct option is C

326040

Method 1
There are four possible cases that satisfy the condition of forming 4-letter passwords with at least one symmetric letter.
Case 1: 1 symmetric letter + 3 unsymmetric letters
Case 2: 2 symmetric letters + 2 unsymmetric letters
Case 3: 3 symmetric letters + 1 unsymmetric letter
Case 4: All 4 symmetric letters
= {(${}^{11}$C${}_1$ $\times$ ${}^{15}$C${}_3$) + (${}^{11}$C${}_2$ $\times$ ${}^{15}$C${}_2$) + (${}^{11}$C${}_3$ $\times$ ${}^{15}$C${}_1$) + (${}^{11}$C${}_4$) }
=5005 + 5775 + 2475 + 330
=13585 $\times$ 4! = 326040
Method 2
Total number of cases- cases where there are no symmetric letters

${}^{26}$C${}_4$ -${}^{15}$C${}_4$ = 13585 $\times$ 4! = 326040