Permutations and Combinations — Class 11 Mathematics

This chapter develops counting techniques. The fundamental principle of counting (multiplication principle) states that if one event can occur in ways and a following event in ways, the two together occur in ways. A permutation is an arrangement in a definite order; the number of permutations of distinct objects taken at a time is , where and . When repetition is allowed there are arrangements, and when some objects repeat ( of each kind) the count is . A combination is a selection where order does not matter; the number of combinations is . Key relations include , , and the Pascal rule . Selecting objects out of is the same as rejecting the remaining , which explains the symmetry of . These ideas count three-digit numbers, signals from flags, words formed from the letters of a given word, committees with conditions, card hands drawn from a deck, and arrangements where some objects must stay together or apart.