Let S={1,2,3,4,5,6} (a) How many subsets are there of...

Let $S=\{1,2,3,4,5,6\}$
(a) How many subsets are there of cardinality 4 ?
(b) How many subsets of cardinality 4 have \{2,3,5\} as a subset?
(c) How many subsets of cardinality 4 contain at least one odd number?
(d) How many subsets of cardinality 4 contain exactly one even number?

Key Concepts

Counting with Exact Conditions

Counting with exact conditions requires carefully selecting and counting the elements that fit precise criteria, such as having exactly one element belonging to a specific category (for example, even numbers). This involves partitioning the problem into parts where the restricted criteria are met exactly, ensuring an accurate count under strict constraints.

Conditional Subset Counting

Conditional subset counting involves determining the number of subsets that not only have a specific size but also satisfy additional conditions, such as containing a particular fixed subset or having a prescribed number of elements from certain categories (like even or odd numbers). This technique helps address restrictions applied to the subsets being counted.

Complement Principle

The complement principle is a useful counting technique where, instead of counting the desired cases directly, one counts the total number of possibilities and then subtracts the number of cases that do not meet the given condition. This approach is particularly helpful for questions involving 'at least' scenarios, such as requiring at least one odd number in a subset.

Combinatorics

Combinatorics is the area of mathematics concerned with counting, arranging, and combining objects. In the context of this problem, it provides the framework for determining the number of subsets of a given set that meet particular criteria.

Binomial Coefficient

The binomial coefficient, often denoted as 'n choose k', represents the number of ways to choose k elements from a set of n distinct elements without regard to the order of selection. It is fundamental for calculating the basic number of subsets of a fixed size in any given set.

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