Permutations and combinations Binomial coefficients An ordered Set a 1 , a 2 ,…, a r of r distinct objects selected from a set of n objects is called a permutation of n things taken r at a time. The number of permutations is given by n P n = n ( n − 1)( n − 2)⋯ ( n − r + 1). When r = n , the number n P r = n ( n − 1)( n − 2)⋯ is simply the number of ways of arranging n distinct things in a row. This expression is called the factorial n and is denoted by n !. It follows that n P r = n !/( n − r )!. By convention 0! = 1. Combinatorics and Pascal’s Triangle Let’s calculate some values of n C r . We start with 0 C 0. Then we find 1 C 0 and 1 C 1. Next, 2 C 0, 2 C 1 and 2 C 2. Then 3 C 0, 3 C 1, 3 C 2 and 3 C 3. We can write down all these results in a table: 0 C 0 = 1 1 C 0 = 1 1 C 1 = 1 2 C 0 = 1 2 C 1 = 2 2 C 2 = 1 3 C 0 = 1 3 C 1 = 3 3 C 2 = 3 3 C 3 = 1 4 C 0 = 1 4 C 1 = 4 4 C 2 = 6 4 C 3 = 4 4 C 4 = 1 5 C 0 =