“A problem is a chance for you to do your best.” – Duke Ellington
Fundamental principle of counting If an event can occur in m different ways, following which another event can occur in n different ways, then the total number of occurrence of the events in the given order is m × n.
The number of permutations of n different things taken r at a time, where repetition is not allowed, is denoted by nPr and is given by \( {^n}P_r = \frac{n!}{(n-r)!}\), where 0 ≤ r ≤ n.
n! = 1 × 2 × 3 × …×n
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n! = n × (n – 1) !
The number of permutations of n different things, taken r at a time, where repetition is allowed, is nr
The number of permutations of n objects taken all at a time, where p1 objects are of first kind, p2 objects are of the second kind, …, pk objects are of the kth kind and rest, if any, are all different is \( \frac{n!}{{p_1}! {p_2}! … {p_k}!} \)
The number of permutations of n different things taken r at a time, where repetition is not allowed, is denoted by nCr and is given by \( {^n}C_r = \frac{n!}{r! (n-r)!}\), where 0 ≤ r ≤ n.
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