• Overview
• Details
• Exercises

Counting techniques are foundational in combinatorics and probability. We cover the product rule, permutations, combinations, and variations with/without repetition.

• Product and sum rules; inclusion–exclusion (preview).
• Permutations without repetition: n!; with k selected: nPk = n!/(n-k)!.
• Combinations without repetition: nCk = \binom{n}{k} = n!/(k!(n-k)!).
• With repetition allowed: stars and bars — combinations with repetition: \binom{n+k-1}{k}.
• Multisets and multinomial coefficients: \frac{n!}{n_1!n_2!\cdots n_m!}.
• Ordering vs selection: when order matters (permutation) vs not (combination).
• Applications: passwords, seating arrangements, lottery probabilities.

1. How many 8-character passwords from 26 lowercase letters with no repetition? With repetition?
2. In how many ways can 5 people be seated around a round table? (Consider rotational symmetry.)
3. Using stars and bars, count the number of nonnegative integer solutions to x_1 + x_2 + x_3 = 10.