Write each as a single factorial. i) 56*6! i) (n + 3)(n + 2)(n + 1)! i) (n² + n)(n - 1)!

Q: Prove by mathematical induction that the sum of cubes of three consequtive positive integers n is…

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Q: FACTORIAL The number of different ways to arrange n objects is given by n!= nx (n-1)x(n-2)x...x3x2x1…

A: Given: The number of different ways to arrange n objects is given by  n! = n×n-1×n-2×...×3×2×1

Q: Given that the expression (n²+n+1) n! can be written as a(n + 2)! + b(n + 1)! + cn! where a, b, c…

A: The main objective is to find the sum value.

Q: Examine the properties of the operation o defined by the equation xoy= x2+y2 in the N natural…

A: Given: The equation x∘y=x2+y2 in the N natural numbers set

Q: Prove that N × N × N is countable. Does your argument generalize to the Cartesian product of k…

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Q: Show that for every integer n, n³ + 2 is not divisible by 4.

A: Here, we use rules of divisibility.

Q: 17. Expand using Pascal's Triangle or the Binomial Theorem. Be sure to show the setup. Then,…

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Q: (n+1)! (n-1)! = 10n

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Q: A man pays $160 a year for a life insurance policy with coverage of $145,000 payable to his survivor…

A: According to given data: The probability that the he will live through the year is 0.9992. The…

Q: 2. 6n - 1 is divisible by 5 for n any positive integer

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Q: Show that for positive integer n,

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Q: Given positive odd integer (a), find all possible divisors, which are always divided a + (a + 1)° +…

A: Odd numbers are those numbers which are not divisible by 2. For example: 3, 5, 7,.... Those numbers…

Q: let A {a, b} and B = {4, 5, 6}. %3D

A: A=a,bB=4,5,6 To find : a A×Ab B×B

Q: Write each as a single factorial. i) 90*8! ii) (n² + n)(n 1)! iii) (n +1)!/(n -1)!

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Q: 3) (n² + 10n+18) ÷ (n+ 5)

A: To divide the following expression.n2+10n+18÷n+5

Q: Expand the following using the Binomial Theorem or Pascal's Triangle: (x- 2)* a. x*-8x3 + 32x + 16…

A: Binomial expansion formula (a+b)n=C0nan+C1nan-1b+C2nan-2b2.......................Cnnbn

Q: Show that for any integer n , n2 – 2 is not divisible by 3.

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Q: Construct the seventh and eighth rows of Pascal’s tri-angle and write the binomial expansions of (x…

A: Pascals triangle: 11 11 2 11 3 3 11 4 6 4 11 5 10 10 5 11 6 15 20 15 6 11 7 21 35 35 21 7 1

Q: Show that the the greatest common divisor gcd(16" – 4",17" – 2") is divisible by 3 for all n E N.

A: Consider the provided question,

Q: divisible by 3 for

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Q: How many 6-digit numbers have three even and three odd digits? (The first digit cannot be zero.)

A: It is given that a 6-digit number is to be formed which will have three even and three odd digits.…

Q: Disprove  For any integer n, n2 + 5 is not divisible by 5.

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Q: Examine the properties of the operation o defined by the equation xoy= x2+y2 in the N natural…

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Q: Can you assist  me in  verifing    that    for    all    n ≥ 1,    the    sum    of    the    first…

A: Here, we have to verify that ∀n≥1, the sum of the squares of the first 2n positive integers is given…

Q: 1+1+5215+5102

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