Consider the following definitions for sets of characters: Digits = {0, 1, 2, 3,

Question

Consider the following definitions for sets of characters: Digits = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} Letters = {a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, y, z} Special characters = {*, &, $, #} Compute the number of passwords that satisfy the given constraints. Strings of length 6. Characters can be special characters, digits, or letters. Strings of length 7, 8, or 9. Characters can be special characters, digits, letters. Strings of length 7, 8, or 9. Characters can be special characters, digits, letters. The first character cannot be a letter.

Explanation / Answer

Answer

a

Assuming repetition is allowed:

1st position of the password has 10 + 26 + 4 = 40 choices

similarly, 2nd, 3rd, 4th, 5th and 6th position has 40 choices each

so, number of possible strings is 406

now, the 6 positions can be arranged in 6! way

So the total possibility becomes 406 * 6!

b.

Taking the understanding from (a)

String for length 7 will have total number of possiblilities = 407 * 7!

String for length 8 will have total number of possiblilities = 408 * 8!

String for length 7 will have total number of possiblilities = 409 * 9!

And therefore, total number of possible strings with length 7, 8 or 9 = (407 * 7!) + (408 * 8!) + (409 * 9!)

c.

For string of length 7:

1st position of the password has 10 + 4 = 14 choices

2nd, 3rd, 4th, 5th, 6th and 7th position has 40 choices each

so, number of possible strings is 14 * 406

now, the 7 positions except the 1st position can be arranged in 6! way

So the total possibility becomes (14* 406 * 6!)

Similarly, for stings of length 8, the total possibility becomes (14* 407 * 7!)

and for strings of length 9, the total possibility becomes (14* 408 * 8!)

Therefore, total number of possible strings with length 7, 8 or 9 = (14* 406 * 6!) + (14* 407 * 7!) + (14* 408 * 8!)

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