A few examples of what you can ask Wolfram|Alpha about:
Factorials and Combinations
do computations with factorials
compute binomial coefficients (combinations)
compute a multinomial coefficient
evaluate a double factorial binomial coefficient
Integer Partitions
compute partitions of an integer
specify a constraint on the number of parts
restrict to partitions into distinct parts
compute the number of partitions
Integer Compositions
compute compositions of an integer
specify a constraint on the parts
Enumeration Problems
count the number of possible colored bracelets
count the number of possible fixed necklaces
count Lyndon words
Combinatorial Functions
compute a Bernoulli number
compute a Stirling number
compute a Frobenius number
compute Catalan numbers
compute Clebsch-Gordan coefficients
compute Wigner coefficients
Latin Squares
get information about Latin squares
compute the number of Latin squares of a specified size
count normalized Latin squares
compute bounds on the number of large-size Latin squares