Master the concept of Permutations for CBSE Class 11 Applied Mathematics. Learn formulas, factorial notation, and solve step-by-step arrangement problems.
If an event can occur in m ways and a second independent event can occur in n ways, then the sequence of two events can occur in m × n ways. This principle forms the basis for permutations, which are ordered arrangements of objects. For example, if you have 3 shirts and 2 trousers, you can form 3 × 2 = 6 distinct outfits.
Step 1: Identify the number of positions (3).
Step 2: Apply the multiplication principle: 3 options for the first, 2 for the second, 1 for the third.
Step 3: Calculate 3 × 2 × 1 = 6.
Answer: 6 ways.
The product of the first n natural numbers is denoted by n!. It represents the number of ways to arrange n distinct objects in a row. By definition, 0! = 1 and 1! = 1.
Step 1: Write the expansion: 5 × 4 × 3 × 2 × 1.
Step 2: Multiply: 5 × 4 = 20; 20 × 3 = 60; 60 × 2 = 120.
Answer: 120
The number of permutations of n distinct objects taken r at a time is denoted by P(n, r). This formula accounts for the order of selection, which is critical in arrangement problems.
Step 1: Substitute n=6, r=2 into the formula: 6! / (6−2)!.
Step 2: Simplify: 6! / 4! = (6 × 5 × 4!) / 4!.
Step 3: Cancel 4!: 6 × 5 = 30.
Answer: 30