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- Imagine you have 331 sheep, and you are going to use primitive counting to write a tally stick for the King. Suppose you carry out the procedure as usual, filling the pens with sheep, and liberating the sheep from pen 2 (and any extra sheep); then doing it again, doing it again, etc., until you get down to one sheep.How many sheep will you have in the first pen After one round? After two rounds? After three rounds?Let l be a line in the x-yplane. If l is a vertical line, its equation is x = a for some real number a. Suppose l is not a vertical line and its slope is m. Then the equation of l is y = mx + b, where b is the y-intercept. If l passes through the point (x₀, y₀), the equation of l can be written as y - y₀ = m(x - x₀). If (x₁, y₁) and (x₂, y₂) are two points in the x-y plane and x₁ ≠ x₂, the slope of line passing through these points is m = (y₂ - y₁)/(x₂ - x₁). Instructions Write a program that prompts the user for two points in the x-y plane. Input should be entered in the following order: Input x₁ Input y₁ Input x₂Let A be the set of all strings of decimal digits of length 5. For example 24157 and05189 are strings in A.(a) How many strings in A have exactly one 7?(b) How many strings in A have at least two 2’s?(c) How many strings in A have the digits in a strictly decreasing order? For example97643 and 54321 are such strings, but 14820 and 95421 are not.
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- 1: Given a fixed integer B (B ≥ 2), we demonstrate that any integer N (N ≥ 0) can bewritten in a unique way in the form of the sum of p+1 terms as follows:N = a0 + a1×B + a2×B2 + … + ap×Bpwhere all ai, for 0 ≤ i ≤ p, are integer such that 0 ≤ ai ≤ B-1.The notation apap-1…a0 is called the representation of N in base B. Notice that a0 is theremainder of the Euclidean division of N by B. If Q is the quotient, a1 is the remainder of theEuclidean division of Q by B, etc.1. Write an algorithm that generates the representation of N in base B. 22. Compute the time complexity of your algorithm.Let :P: Birds can fly.Q: 2+1=4.R: x is an integer.S: π is a rational number. ~ (p ⊕ ~ q) ν ~ (~r Λ s) GIVEN THE FOLLOWING PROPOSITIONS A. ILLUSTRATE THE COMBINATORIAL CIRCUIT.The binomial coefficient C(N,k) can be defined recursively as follows: C(N,0) = 1, C(N,N) = 1, and for 0 < k < N, C(N,k) = C(N-1,k) + C(N - 1,k - 1). Write a function and give an analysis of the running time to compute the binomial coefficients as follows: A. The function is written using dynamic programming.
- Given f(x)=x2+6x and g(x)=1−x2, find f+g, f−g, fg, and fg. Enclose numerators and denominators in parentheses. For example, (a−b)/(1+n).A) Given a real number x and a positive integer k, determine the number of multiplications used to find x2k starting with x and successively squaring (to find x2, x4, and so on). Is this a more efficient way to find x2k than by multiplying x by itself the appropriate number of times?Given g = {(1,c),(2,a),(3,d)}, a function from X = {1,2,3} to Y = {a,b,c,d}, and f = {(a,r),(b,p),(c,δ),(d,r)}, a function from Y to Z = {p, β, r, δ}, write f o g as a set of ordered pairs.