Permutations and Combinations

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If an event can occur in m different ways, following which another event can occur in n different ways, then the total number of occurrences of the events in the given order is m × n. This principle c…

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Using the fundamental principle of counting: 4 × 3 = 12 ways. For each shirt choice (4 ways), there are 3 pant choices, giving us 4 × 3 = 12 different combinations.

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A permutation is an arrangement in a definite order of a number of objects taken some or all at a time. In permutations, the order of arrangement matters. For example, ABC and BAC are different permut…

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n! represents the product of first n natural numbers. n! = 1 × 2 × 3 × ... × (n-1) × n. Examples: 3! = 1 × 2 × 3 = 6, 4! = 1 × 2 × 3 × 4 = 24. By definition, 0! = 1.

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nPr = n!/(n-r)!, where 0 ≤ r ≤ n. This gives the number of ways to arrange r objects from n distinct objects where order matters and repetition is not allowed.

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5P3 = 5!/(5-3)! = 5!/2! = (5 × 4 × 3 × 2 × 1)/(2 × 1) = 120/2 = 60. Alternatively, 5P3 = 5 × 4 × 3 = 60.

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Since all letters are distinct and we use all 4 letters, the answer is 4P4 = 4! = 24 words. Each arrangement like ROSE, REOS, SORE, etc., is counted as a different word.

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When repetition is allowed, the number of permutations of n different objects taken r at a time is nr. Each position can be filled in n ways independently.