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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Mathematics

Basic Definitions Related to Probability

Total Number of Combinations

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

$\frac{5}{11}$

$\frac{6}{11}$

$\frac{1}{2}$

$\frac{36}{55}$

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