Quick Elementary Combinatorics Question / Verification

[tylerc1991]

Homework Statement

I am using a book that doesn't have any solutions in it, so I would like to be sure that I am doing the problems right before I move on. The question is below:

In a programming language, a variable name must start with a letter or the underscore character, and succeeding characters must be letters, digits, or the underscore character. Uppercase and lowercase letters are considered different.
(a) How many variable names with exactly 5 characters can be formed?
(b) How many are there with at most 5 characters?
(c) How many are there with at most 5 characters, if they must read the same forwards and backwards? (note that kayak is acceptable but Kayak is not).

The Attempt at a Solution

(a) There are 53 possible characters for the first position, and 63 possibilities for the remaining 4 positions. So there are
$53 \cdot 63^4 = 834,906,933$
possible 5 character variable names.

(b) For some $k$, there are $53 \cdot 63^{k-1}$ possible variable names of length $k$. So the number of possible variable names with length at most 5 is
$\sum_{k=1}^5 53 \cdot 63^{k-1} = 848,373,173$ possible variable names with length at most 5.

(c) For length 1, there are only 53 such possible variable names. For length 2, the first and last characters must be the same, so there are again only 53 such possible variable names. For length 3, there are 53 * 63 variable names, and the same number of variable names of length 4. For length 5, there are 53 * 63 * 63 possible variable names. Hence, there are
$53 + 53 + 53 \cdot 63 + 53 \cdot 63 + 53 \cdot 63^2 = 217,141$
possible variable names with the desired property.

Thanks!

 

[LCKurtz]

Homework Statement

I am using a book that doesn't have any solutions in it, so I would like to be sure that I am doing the problems right before I move on. The question is below:

In a programming language, a variable name must start with a letter or the underscore character, and succeeding characters must be letters, digits, or the underscore character. Uppercase and lowercase letters are considered different.
(a) How many variable names with exactly 5 characters can be formed?
(b) How many are there with at most 5 characters?
(c) How many are there with at most 5 characters, if they must read the same forwards and backwards? (note that kayak is acceptable but Kayak is not).

The Attempt at a Solution

(a) There are 53 possible characters for the first position, and 63 possibilities for the remaining 4 positions. So there are
$53 \cdot 63^4 = 834,906,933$
possible 5 character variable names.

(b) For some $k$, there are $53 \cdot 63^{k-1}$ possible variable names of length $k$. So the number of possible variable names with length at most 5 is
$\sum_{k=1}^5 53 \cdot 63^{k-1} = 848,373,173$ possible variable names with length at most 5.

(c) For length 1, there are only 53 such possible variable names. For length 2, the first and last characters must be the same, so there are again only 53 such possible variable names. For length 3, there are 53 * 63 variable names, and the same number of variable names of length 4. For length 5, there are 53 * 63 * 63 possible variable names. Hence, there are
$53 + 53 + 53 \cdot 63 + 53 \cdot 63 + 53 \cdot 63^2 = 217,141$
possible variable names with the desired property.

Thanks!

Your calculations look correct to me.