Counting & Binomials
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Pascal's Triangle and nCr
Factorials
Factorials are shortcut used to express decreasing products of integers such as
The definition is
Fractions of factorials as partial products
Binomial Coefficient nCr
The number of ways to choose r items from a set of n items can be expressed as (nr), nCr or nCr. The number of combinations can be calculated using the formula
Pascal's Triangle
Pascal's triangle is a triangular array where each number is the sum of the two directly above it, beautifully revealing the coefficients of binomial expansions. Each term corresponds to a specific value of nCr. Its symmetry and simple construction make it a powerful tool for exploring combinatorial relationships and probability.
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Binomial Theorem
The expansion of (a+b)ⁿ
The binomial theorem allows us expand expressions of the form (a+b)n:
or in summation form:
Counting
Counting principles
If we have n ways to do one thing and k to do another, then there are n×k ways to do BOTH (first thing AND the second thing), and n+k ways to do one thing OR the other thing.
Arrangements WITH repetition are k^n
In any scenario where we can put k items into each of n positions, we have kn possible arrangements.
Ordering items
The number of different orders in which n items can be arranged is
where each box is a "slot" in the order. In the first slot we can put any of n items, in the second any of the n−1 remaining items, and so on. We then multiply all of these together to get n!
Permutations
The number of permutations, defined as an ordered arrangement, of r items from a set of n items can be calculated by:
Combinations
Combinations are arrangements of items where order does not matter. For combinations of r items from a set of n:
which is the binomial coefficient.
Counting with Restrictions
Counting with subcases
When counting scenarios that have distinct "sub-cases", we count each case separately, and add the results.
Orderings where certain items need to stay together
When counting arrangements where certain items need to stay together, we treat those items as one item, and calculate the number of permutations. Then, we multiply by the number of ways the group can be internally ordered.
For example, if we have A, B, C and D but A, B & C must be together, then we have
But since [ABC] can be internally ordered 3=6 ways, we have 12 total arrangements
Orderings where certain items need to stay apart
When counting arrangements where certain items need to stay apart, we first consider the number of ways to arrange the unrestricted items. Then we imagine inserting the remaining letters into different "slots" between the regular items.
For example, if we have A, B, C and D but A, B cannot be adjacent to each other, then we have
of the unrestricted letters. Then A can go in any of the 3 slots, then B in any of the remaining 2 which makes 3×2=6 ways to inert A and B. Thus there are a total of 2×6=12 arrangements.
Complementary counts
Imagine a scenario that can be split into two cases:
case A can be done in n ways
case B can be done in k ways
The addition of n+k must give the total count for the scenario. This is very useful for solving counting questions, as it allows us to subtract a number of undesirable cases from the number of total cases to find our desired answer.