Can someone help me with this question? I need step by step on how you solved th

Question

Can someone help me with this question? I need step by step on how you solved the problem and I need names of any theorms or formulas you are using please.

A valid password is a string of between 6 and 12 characters, inclusive, where each character can be an uppercase or a lowercase letter (i.e, a, . . . , z, A, ..., Z), a dollar sign ($), an underscore ( ), or a digit (0, ..., 9), except that the first character must be a digit. For example, 0aaaaa is a valid password; but Aaaaaa is not. How many different passwords are there? (Express your answer as a sum.)

Thanks!!

Explanation / Answer

There can be 6 to 12 characters

Lets take for 6 character

So first character must be a digit. , So we have 10 possiblities . [0 to 9]

So 2nd character must be a anything. , So we have 10 + 52 + $ + _ = 64 possiblites

So 3rd character must be a anything. , So we have 10 + 52 + $ + _ = 64 possiblites

So 4th character must be a anything. , So we have 10 + 52 + $ + _ = 64 possiblites

So 4th character must be a anything. , So we have 10 + 52 + $ + _ = 64 possiblites

So 6th character must be a anything. , So we have 10 + 52 + $ + _ = 64 possiblites

So total possiblites for six digits 10*64*64*64*64*64

Similarly we can find for seven digits : 10*64*64*64*64*64 *64

Similarly we can find for eight digits : 10*64*64*64*64*64 *64*64

Similarly we can find for nine digits : 10*64*64*64*64*64 *64*64*64

Similarly we can find for ten digits : 10*64*64*64*64*64 *64*64*64*64

Similarly we can find for eleven digits : 10*64*64*64*64*64 *64*64*64*64*64

Similarly we can find for twelve digits : 10*64*64*64*64*64 *64*64*64*64*64

Summing all the above is our answer

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