How to work out which formula to use

The key issue is to translate the problem into the two slip questions (numbered or blank? with or without repetition?). A few worked examples:

Observation — How to work out which formula to use

• “How many 5-letter words can be formed with the alphabet?” Numbered slips (each letter is distinguishable), with repetition (the letters can be repeated): $26^5$.
• “How many anagrams of the word CASA?” Permutations with repetition (two identical A’s): $\dfrac{4!}{2!} = 12$.
• “How many groups of 3 representatives from a class of 25?” Numbered slips (the students are distinguishable) but the order does not count — hence blank for counting purposes: $\dbinom{25}{3} = 2300$.
• “How many handshakes among 10 people?” It is like choosing 2 people out of 10: $\dbinom{10}{2} = 45$.
• “How many ice-cream cups with 3 scoops chosen among 6 flavours, being allowed to repeat?” Blank with repetition: $\dbinom{6+3-1}{3} = \dbinom{8}{3} = 56$.

Links

Topics: Combinatorics
Concepts: Combinations · Arrangements · Permutations
Methods: Slips table
Skills: Combinatorial calculus · Modelling