Pascal's triangle is a triangular array of numbers where each number is the sum of the two numbers directly above it. This triangle can be used to calculate combinations and permutations, which are ways of counting how many different ways a set of objects can be arranged or chosen.
nCr represents the combination of n objects taken r at a time. In other words, it represents the number of ways you can choose r objects from a set of n objects without regard to the order in which they are chosen.
To find the rᵗʰ term in the nᵗʰ row of Pascal's triangle, you would use the formula nCr = (n-1)C(r-1) + (n-1)C(r). This can be interpreted as taking the sum of the two numbers directly above the desired term in the previous row.
Yes, the nCr formula can be proven using mathematical induction. By starting with the base case of n=1 and r=1, and then showing that the formula holds for the next row and term, we can conclude that it holds for all subsequent rows and terms in Pascal's triangle.
Pascal's triangle can be used in probability to calculate the chances of certain outcomes occurring. For example, if you roll a die 5 times and want to know the probability of getting exactly 3 heads, you can use the nCr formula to find the number of ways this can happen and then divide it by the total number of possible outcomes (6^5). This can also be used in more complex probability problems involving combinations and permutations.