Three randomly chosen non‑negative integers x, y and z are found to satisfy the equation x + y + z = 10. Then the probability that z is even, is

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Question

Three randomly chosen non‑negative integers x, y and z are found to satisfy the equation x + y + z = 10. Then the probability that z is even, is

$\frac{5}{11}$

$\frac{6}{11}$

$\frac{36}{55}$

$\frac{1}{2}$

x + y + z = 10

Total non‑negative solution = ^10+3–1C_3–1 = ¹²C₂ = $\frac{12 \times 11}{2}$ = 66

If z is even, then x + y = 10 or 8 or 6 or 4 or 2 or 0

$∴$ Number of solutions = ¹¹C₁ + ⁹C₁ + ⁷C₁ + ⁵C₁ + ³C₁ + ¹C₁ = 11 + 9 + 7 + 5 + 3 + 1 = 36

$∴$ Probability = $\frac{36}{66} = \frac{6}{11}$ .

We are given that three non-negative integers x, y, z satisfy x + y + z = 10. We need to find the probability that z is even.

Step 1: Total number of solutions

The number of non-negative integer solutions to x + y + z = 10 is given by the combination formula with repetition (stars and bars method):

$Total = C (10 + 3 - 1, 3 - 1) = C (12, 2) = \frac{12 \times 11}{2 \times 1} = 66$

Step 2: Favorable cases where z is even

Let z = 2k, where k is a non-negative integer. Then the equation becomes x + y + 2k = 10, or x + y = 10 - 2k.

For each k, the number of non-negative integer solutions for (x, y) is given by:

$N (k) = C ((10 - 2 k) + 2 - 1, 2 - 1) = C (11 - 2 k, 1) = 11 - 2 k$

k can range from 0 to 5 (since z = 2k ≤ 10). So the total favorable number is the sum over k=0 to 5:

$Favorable = \sum_{k = 0}^{5} (11 - 2 k) = 11 + 9 + 7 + 5 + 3 + 1 = 36$

Step 3: Compute the probability

Probability = Favorable / Total = 36 / 66 = 6 / 11.

So the correct answer is $\frac{6}{11}$ .

Related Topics:

This problem involves combinatorics (counting number of solutions) and probability. Key concepts include:

Stars and bars method for counting non-negative integer solutions to linear equations.
Summation of arithmetic series.
Basic probability calculation.

Formulae:

Number of non-negative integer solutions to x₁ + x₂ + ... + x_r = n is C(n + r - 1, r - 1).

Probability = (Number of favorable outcomes) / (Total number of outcomes).

Detailed Solution