Plz I need help
Let n be a positive integer. Let k be a positive integer.
(a) Using a counting argument, show that C(n - 1, k - 1) + C(n - 1, k) = C(n,k).
(b) Using a counting argument, show that C(n,0) + C(n,1) + C(n,2) + ... + C(n,n) = 2^n.
(c) Using a counting argument, show that C(n,0) + C(n + 1,1) + C(n + 2,2) + C(n + 3,3) + ... + C(n + k,k) = C(n + k + 1,k)
(d) Using a counting argument, show that C(m,0) C(n,r) + C(m,1) C(n,r - 1) + ... + C(m,r) C(n,0) = C(m + n,r).
(e) There are n^2 ordered pairs (a,b) of positive integers, where 1 \le a, b \le n. Using a counting argument, show that this number is also equal to
(n + 1)^2 - 2n - 1
(f) There are n^3 ordered triples (a,b,c) of positive integers, where 1 \le a, b, c \le n. Using a counting argument, show that this number is also equal to
(n + 1)^3 - 3n^2 - 3n - 1
(a) The left-hand side counts the number of ways to choose k elements from a set of n - 1 elements. The first term, C(n - 1, k - 1), counts the number of ways to choose k - 1 elements, and the second term, C(n - 1, k), counts the number of ways to choose k elements. To form the set of k elements in C(n, k), we simply add an element n to each of these two sets. So, C(n - 1, k - 1) + C(n - 1, k) = C(n, k).
(b) The left-hand side counts the number of ways to choose 0 to n elements from a set of n elements. For each positive integer i from 0 to n, C(n, i) counts the number of ways to choose i elements. The sum on the left-hand side is equivalent to counting the number of possible combinations of n elements, which is 2^n.
(c) The left-hand side counts the number of ways to choose 0 to k elements from a set of n + k elements. Each term C(n + i, i) counts the number of ways to choose i elements from a set of n + i elements. The sum on the left-hand side is equivalent to counting the number of possible combinations of n + k elements, which is C(n + k + 1, k).
(d) The left-hand side counts the number of ordered pairs (X, Y) where X is a set of r elements selected from a set of m elements and Y is a set of r elements selected from a set of n elements. For each positive integer i from 0 to r, C(m, i) C(n, r - i) counts the number of ordered pairs where X has i elements and Y has r - i elements. The sum on the left-hand side is equivalent to counting the number of possible combinations of r elements from the set of m + n elements, which is C(m + n, r).
(e) The left-hand side counts the number of ordered pairs (a, b) of positive integers where 1 <= a, b <= n. The expression (n + 1)^2 - 2n - 1 counts the total number of ordered pairs of positive integers where 1 <= a, b <= n + 1 and then subtracts the ordered pairs (n + 1, n + 1) and (n + 1, i) for 1 <= i <= n, as well as (i, n + 1) for 1 <= i <= n. This gives us the total number of ordered pairs of positive integers where 1 <= a, b <= n.
(f) The left-hand side counts the number of ordered triples (a, b, c) of positive integers where 1 <= a, b, c <= n. The expression (n + 1)^3 - 3n^2 - 3n - 1 counts the total number of ordered triples of positive integers where 1 <= a, b, c <= n + 1 and then subtracts the ordered triples (n + 1, n + 1, n + 1), (n + 1, n + 1, i), (n + 1, i, n + 1), and (i, n + 1, n + 1) for 1 <= i <= n, as well as all ordered triples where at least one element is equal to n + 1. This gives us the total number of ordered triples of positive integers where 1 <= a, b, c <= n.
