Give a combinatorial proof that: 2 η η 2η Σ+(%) =n(*1) k=1 [Hint: There are two groups of n members each].

Q: 1. Prove that for any n € Z, 3n² +11n + 15 is odd.

A: To prove that for any integer n, the expression 3n2+11n+15 is odd, it can use the properties of odd…

Q: e²x3 In Expand

A: ln(e2x3/y5) now here use ln(xy) = lnx + lny and ln(x/y) = lnx - lny here apply rule lne2 + lnx3 -…

Q: The number of permutations of n elements taken rat a time is denoted by nPr and is computed by or…

A: The number of permutations of n elements taken r at a time is given by nPr where nPr is given by,

Q: Show that if n>=m then the number of m-cycles in Sn is given by: n(n-1)(n-2)...(n-m+1)/m

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Q: Exercise 8.4. Suppose o e Sym(n) is a k-cycle. Show that TOT-¹ is also a k-cycle for every 7 €…

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Q: 4. Prove or disprove the following statement. U(8) is cyclic. [from #1, 4.5]

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Q: Exercise 5.3.2. Compute the order of each of the following elements in their respective groups. (1)…

A: Using the definition of the various groups given in the question we solve this problem. The detailed…

Q: Suppose a, n E Z and 0 < a<n. Prove that )-(".")-(C ) n- 1 in = a a а — 1 in two different ways:…

A: We have to prove, na = n-1a+n-1a-1 First let us prove this using formula. Using Formula…

Q: Σ1 (3)* (?)² i=1 ►Give a combinatorial proof of the following identity (3η) = (²n) HINT: Remember…

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Q: (1 2 3 4 5 5 1 3 6 2 2. Let o = E S6. 1. Wi des 2. 3. 4. What is the size of the conjugacy class of…

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Q: Prove that 26 Σ j=0 26 (20) (-1)³ = 0. Using Part (a) (or otherwise), show that the number of…

A: We have to prove different quantities.

Q: 1. Let A CU and B C U. Suppose |A n B| = 8, |A| = 10, |B| = 15 and |U| 20. Determine %3D %3D |P(P(AU…

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Q: APPLICATIONS 225 To begin, we note that the gamma function is the solution to the difference…

A: Given equation is  ak+1=kak                         (i) Let us substitute ak=esk in eq. (i). It…

Q: Question 3 (3.1) Use the definition of O(f) to show that n² is O(n² log n). (3.2) Is n²log n also…

A: Question 3(3.1) To show that n^2 is O(n^2 * log(n)), we need to find positive constants c and k such…

Q: (e) n = 0(n²) and n2 = 0(n) (f) cos(n) = 0(1). (g) A tree with n vertices has n – 1 edges. (h) If p…

A: solution e n=On2  and n2=On False  as because   n=On  and n2=On2 f   cosn=O1 True as because we can…

Q: In general, if (12 · · · k) is a cycle of length k and o is any permutation then o(12 ·.· k)o -1 =…

A: Given: T=(12.....k)

Q: Show that E-o (-1)*(")=0, and then verify that () + C) + C)+ - = (") +C) + () +-- -G) + G) + G)-…

A: We can show given identity by definition of Binomial theorem:

Q: . Consider the permutation ♂ € S5 given by o = (1 2 3)(2 4 5) (notice it is NOT expressed as a…

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Q: 2n 2n 1/2n + 2 1, 2 n +1 n -

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Q: 12. Use Mathematical Induction to prove that if the set S has n elements, then P(S) has 2" elements.

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Q: J Establish the formulas. induction below bey mathemotical [b.) I+3+5+.. 7(2n-1)=n for all %3D .....

A: Hey, since there are multiple questions posted, we will answer first question. If you want any…

Q: n m+n ) (₁ - ₁) + - + ( ) ( ) = (";"). (™+"). r > a. in valid

A: