Master the concept of Combinations in CBSE Class 11 Applied Mathematics. Learn formulas, properties, and step-by-step solutions for board exam success.
A combination is a selection of objects from a given set where the order of selection does not matter. Unlike permutations, where arrangement is key, combinations focus solely on the grouping of items. If we have n distinct objects, the number of ways to select r objects is denoted by nCr or C(n, r).
Step 1: Identify n=5 and r=2.
Step 2: Apply formula 5! / (2! · (5 − 2)!).
Step 3: Calculate 120 / (2 · 6) = 120 / 12.
Answer: 10
Combinations follow specific algebraic identities that simplify complex calculations. Key properties include the symmetry property and the Pascal identity. These allow for efficient computation when n or r values are large.
Step 1: Calculate 10C8 = 10! / (8! · 2!) = (10 · 9) / 2 = 45.
Step 2: Calculate 10C2 = 10! / (2! · 8!) = (10 · 9) / 2 = 45.
Answer: Both equal 45.
When selecting items, constraints such as 'at least one' or 'excluding specific items' often arise. These are handled by subtracting the unwanted cases from the total possible combinations. This logical approach is essential for solving word problems.
Step 1: Use Pascal identity (n+1)Cr where n=6, r=3.
Step 2: Result is 7C3.
Step 3: 7C3 = (7 · 6 · 5) / (3 · 2 · 1) = 35.
Answer: 35