Question

a combination lock has 50 numbers from zero to 49, and a combination consists of 4 numbers in a specific order with no repeats. find the probability that the combination consists only of even numbers. (round your answer to three decimal places.) the probability that the combination consists only of even numbers is

Explanation:

Step1: Calculate the total number of combinations

The total number of ways to choose 4 non - repeating numbers from 50 numbers is given by the permutation formula \(P(n,r)=\frac{n!}{(n - r)!}\), where \(n = 50\) and \(r=4\).

$$P(50,4)=\frac{50!}{(50 - 4)!}=\frac{50!}{46!}=50\times49\times48\times47=5527200$$

Step2: Calculate the number of combinations with only even numbers

There are 25 even numbers from 0 to 49. The number of ways to choose 4 non - repeating even numbers is also given by the permutation formula with \(n = 25\) and \(r = 4\)

$$P(25,4)=\frac{25!}{(25 - 4)!}=\frac{25!}{21!}=25\times24\times23\times22 = 303600$$

Step3: Calculate the probability

The probability \(P\) is the number of favorable outcomes (combinations with only even numbers) divided by the number of total outcomes (all combinations).

$$P=\frac{303600}{5527200}\approx0.055$$

Answer:

\(0.055\)