Binomial Coefficients: Selected Exercises Preliminaries What is the
Binomial Coefficients: Selected Exercises Preliminaries What is the coefficient of x2y in (x + y)3? (x + y)3 = (x + y)(x + y)(x + y) = (xx + xy + yx + yy)(x + y) = xxx + xyx + yxx + yyx + xxy + xyy + yxy + yyy = x3 + 3x2y + 3xy2 + y3. The answer thus is 3.
There are 8 terms in the formal expansion. Answer: The # of ways to pick the y position in the formal expansion: C(3, 1). 2 Preliminaries How many terms are there in the formal expansion of (x + y)n? How many such terms have exactly 3 ys? This is the coefficient of xn-3y3 in (x + y)n? How many such terms have exactly j ys?
3 The Binomial Theorem Let x & y be variables, and n NN. Partition the set of 2n terms of the formal expansion of (x + y)n into n + 1 classes according to the # of ys in the term: (x + y)n = j=0 to n C( n, j )xn-jyj = C(n, 0)xny0 + C(n,1)xn-1y1 + + C(n, j)xn-jyj + + C(n,n)x0yn. 4
Pascals Identity Let n & k be positive integers, with n > k. Then C(n, k) = C(n - 1, k 1) + C(n - 1, k). Proof by a combinatorial argument: 1. The left hand side counts the number of subsets of size k from a set of n elements. 2. The right hand side counts the same subsets using the sum rule: Partition the subsets into 2 disjoint classes 1. Subsets of k elements that include element 1: 1. Pick element 1: 1 2. Pick the remaining k 1 elements from the remaining n - 1 elements:
C(n - 1, k 1). 2. Subsets of k elements that exclude element 1: 1. Pick the k elements from the n - 1 remaining elements: C(n - 1, k). 5 *30 Give a combinatorial proof that k=1,n kC(n, k)2 = nC( 2n 1, n 1 ). Hint: Show that the equations LHS and RHS are different ways to count the same thing:
The # of ways to select a committee with n members from a group of n math professors & n computer science professors, such that the committee chair is a mathematics professor. 6 *30 Solution Proof by a combinatorial argument The RHS counts the # of such committees by: 1. Pick the chair from the n math professors: n 2. Pick the remaining n 1 members from the remaining 2n 1 professors: C( 2n 1, n 1 )
The LHS counts the same committees by: Partition the set of such committees into disjoint subsets, according to, k, the # of math professors on the committee. For each k, 1. Pick the k math professor members: C(n, k) 2. Pick the committee chair: k 3. Pick the n - k CS professor members: C( n, n k ) = C( n, k ) 7 Combinatorial Identities
Manipulation of the Binomial Theorem Committee arguments Block walking arguments for identities involving sums 8 Manipulation of the Binomial Theorem Prove that C(n, 0) + C(n, 1) + . N. N. N + C(n, n) = 2n. In general,
1. Algebraically manipulate the binomial theorem. 2. Evaluate the resultant equation for suitable values of x & y to get the desired result. Prove that n2n 1 = 1C(n, 1) + 2C(n, 2) + 3C(n, 3) + . N. N. N + nC(n, n). 9
Committee Arguments Show that 1. C(n, k)C(k, m) = C(n, m)C(n m, k m). Hint: committees of k people, m of whom are leaders. 2. k = 0 to r C(m,k)C(n, r - k) = C(m + n, r). Hint: committees of r people taken from m men & n women. 10
Block-Walking Arguments 1. Draw Pascals triangle. 2. Interpret a node in the triangle as the # of ways to walk from the apex to the node, always going down. Show that 1. C(n, k) = C(n 1, k) + C(n 1, k 1) 2. C(n,0)2 + C(n,1)2 + . . . + C(n,n)2 = C(2n,n). 11 Characters
N N . ~ N N N N N N N N N N N N N N N N N
12 *10 Give a formula for the coefficient of xk in the expansion of (x + 1/x)100, where k is an even integer. 13 *10 Solution By the Binomial Theorem, (x + 1/x)100 = j=0 to 100 C(100, j)x100-j(1/x)j
= j=0 to 100 C(100, j)x100-2j. We want the coefficient of x100-2j, where k = 100 2j j = (100 k)/2. The coefficient we seek is C(100, (100 k)/2). 14 20 Suppose that k & n are integers with 1 k < n. Prove the hexagon identity
C(n - 1, k 1)C(n, k + 1)C(n + 1, k) = C(n-1, k)C(n, k-1)C(n+1, k+1), which relates terms in Pascals triangle that form a hexagon. Hint: Use straight algebra. 15 20 Solution C( n 1, k 1 )C( n, k + 1 )C( n + 1, k ) = (n 1)! n! (n+1)! _______________________________________
(k 1)!(n k)! (k + 1)!(n k 1)! k! (n + 1 k)! = C( n 1, k )C( n, k 1 )C( n + 1, k + 1 ). 16
Steps for creating an outline/draft, 4. Lastly, create the bibliography page, also known as the Works Cited Page. You do this by copying and pasting the entire bibliography card for all the details that you used in the paper.
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