Algebra Permutations Mock Tests

The number of non-negative integer solutions to x + y + z = 15 with x ≥ 2, y ≥ 1, and z ≥ 0 is:

A committee of 6 people is to be formed from 8 men and 7 women. If the committee must have at least 3 men and at least 3 women, and the oldest man refuses to serve if the oldest woman is also on the committee, then the total number of ways to form the committee is:

Let alpha, beta, gamma be the roots of x^3 - 6x^2 + 11x - 6 = 0. The value of alpha^3 + beta^3 + gamma^3 is

The number of ways to arrange 6 distinct people around a circular table such that two specific persons A and B always sit together is:

Let f(x) = x^4 + ax^3 + bx^2 + cx + d be a polynomial with real coefficients. If sqrt(3) + sqrt(2) is a root of f(x) = 0 and f(1) = -9, then the sum of the fourth powers of all the roots of f(x) = 0 is equal to:

Let α, β, γ, δ be the roots of the polynomial x⁴ - 10x³ + 35x² - 50x + 24 = 0. If Sₙ = αⁿ + βⁿ + γⁿ + δⁿ, then the value of S₃ is:

The number of ways to choose 3 numbers from {1, 2, 3, ..., 10} such that no two are consecutive is:

Let α, β, γ be the roots of x³ - 6x² + 11x - 6 = 0. The value of 1/α + 1/β + 1/γ equals: