how does pascals triangle relate to permutation and combination?
First read up the section on
"Pascal's triangle and combinations"
in the following article:
http://en.wikipedia.org/wiki/Pascal%27s_triangle
In fact, the whole article makes interesting reading if this is the subject being taught in your class.
Pascal's triangle is a triangular array of numbers that has a lot of mathematical properties. One of its main applications is in calculating combinations and permutations.
To understand the relationship between Pascal's triangle and permutations/combinations, we need to understand the concept of binomial coefficients. The numbers in Pascal's triangle are actually the binomial coefficients, which represent the coefficients of the terms when expanding a binomial expression raised to a power.
Now, let's see how Pascal's triangle can help us with permutations and combinations:
1. Combinations: In combinatorics, a combination is a selection of items from a larger set without considering the order of the items. The number of possible combinations can be determined using binomial coefficients.
For example, let's say we have a set of n items, and we want to select r items from that set. The number of combinations, denoted by C(n, r), can be found in Pascal's triangle by looking at the value in the (n+1)th row and the (r+1)th column. For instance, C(5, 2) = 10, which can be found in the 6th row, 3rd column of Pascal's triangle.
2. Permutations: In combinatorics, a permutation is an arrangement of items from a set in a specific order. The number of possible permutations can also be determined using binomial coefficients and Pascal's triangle.
For example, let's say we have a set of n items, and we want to arrange all the items in a specific order, without repetition. The number of permutations, denoted by P(n), can be found in Pascal's triangle by adding up all the values in the (n+1)th row. For instance, P(4) = 1 + 4 + 6 + 4 + 1 = 16, which is the sum of the values in the 5th row of Pascal's triangle.
In summary, Pascal's triangle provides a convenient way to determine the number of combinations and permutations of a given set of items. By looking at the appropriate row and column in the triangle, we can find the corresponding binomial coefficient and use it in our calculations.