Three randomly chosen non-negative integers x, y and z are found to satisfy the equation x+y+z=1

step 1 in this problem first thing to note that is the total number of ways of dividing n identical amo

Three randomly chosen non-negative integers x, y and z are...

Three randomly chosen non-negative integers $\mathrm{x}, \mathrm{y}$ and $\mathrm{z}$ are found to satisfy the equation $\mathrm{x}+\mathrm{y}+\mathrm{z}=10$. Then the probability that $\mathrm{z}$ is even, is [Adv. 2017] (a) $\frac{36}{55}$ (b) $\frac{6}{11}$ (c) $\frac{1}{2}$ (d) $\frac{5}{11}$

Three randomly chosen non-negative integers $\mathrm{x}, \mathrm{y}$ and $\mathrm{z}$ are found to satisfy the equation $\mathrm{x}+\mathrm{y}+\mathrm{z}=10$. Then the probability that $\mathrm{z}$ is even, is [Adv. 2017]
(a) $\frac{36}{55}$
(b) $\frac{6}{11}$
(c) $\frac{1}{2}$
(d) $\frac{5}{11}$

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Key Concepts

Parity

Parity refers to whether an integer is even or odd, which is a fundamental concept in number theory. It helps in analyzing problems where the constraints involve even and odd numbers, and is particularly useful in problems with conditions or restrictions based on the remainder modulo 2.

Combinatorial Probability

In combinatorial probability, probabilities are calculated as the ratio of the count of favorable outcomes to the total number of possible outcomes. This approach is essential when dealing with discrete sample spaces, especially in problems where counting techniques such as stars and bars are applied to enumerate outcomes.

Stars and Bars

This concept is a counting technique used to determine the number of non-negative integer solutions to equations of the form x1 + x2 + ... + xn = k. It involves representing the equation as a sequence of stars (representing units) and bars (dividing the units among the variables), allowing one to count the number of valid distributions without overcounting or omission.

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