The factorial of n is the product of all positive integers less than or equal to n. It is represented as n!   An example with n = 8 is 8! = 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1. With that in mind, can you find three different sets of numbers (a, b, c) such that a! × b! = c! and a < b < c < 25?

Q: Evaluate if x=4,8,3,-2 ∑x ∑x+1 ∑(x+1) 3∑x - 4

A: Given that  if x=4,8,3,-2

Q: Find each (n+3)(n+2) (4n 5)(3n + 2) (n+7)(n + 5)

A: Since you have posted multiple subparts we solve the first three subparts for you to get the…

Q: What is the sum 1+2+4+8+...+512?

A:

Q: Prove: The sum of an odd number of odd numbers must be odd.

A: Let us prove this by induction, that if n is odd and a1,a2,. . .. .an are odd, then ∑i=1nai is odd.…

Q: Find the smallest number n for which n! is larger than the population of the world.

A: World population is 7.98 billion till October 2022 It according to worldometer by United States.…

Q: How many positive integers less than 1,000,000 have a digit sum of 5?

A: We have to find the number of positive integers 1,000,000 have a digit sum of 5. The number less…

Q: How many triples of integers from 2 through 16 can be formed in which the integers are in increasing…

A: Any triple of integers (i, j, k) with 2 ≤ i ≤ j ≤ k ≤ 16 can berepresented as a string of 15 – 1…

Q: Prove that a three-digit number is a multiple of 9 whenever the sum of the digits is a multiple of…

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Q: How do we proof that 2logn = n?

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Q: For all integers n, if 6 | n, then 9 | ((n^2) + 21n/2)

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Q: Prove a number is divisible by 3 if and only if the sum of its digits is divisible by 3.

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Q: 26 + 4 × [9 - (1 + 4)

A: Given that  26 + 4 × [9 - (1 + 4)] To simplify the expression

Counting Principles

In statistics, the counting principle can be said to be the concept in which the quantity of an item is determined. There can be a single item, or a group of items, or items that are related to each other. In every possible set of items, the quantity of the possible item groups or their sorting measures can be determined with counting principles.