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Chapter 1: Problem 14

Three distinct integers are chosen at random from the first 20 positive integers. Compute the probability that: (a) their sum is even; (b) their product is even.

Short Answer

Expert verified
The probability that their sum is even is calculated using the combinations formula and dividing favourable outcomes by total outcomes. Similar calculation is done for their product being even.

Step by step solution

01

Calculate Total Combinations

Firstly, find out the total combinations that can be formed by selecting 3 integers out of 20. We can use the combination formula for this, which is \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \). Here \( n = 20 \) and \( r = 3 \), so total combinations = \( \binom{20}{3} \).
02

Calculate Favourable Combinations for Part (a)

Now, calculate the favourable outcomes for the sum being even. The sum of three numbers will be even if all three numbers are even, or two are odd and one is even. First, find the combinations of all three being even. These can be obtained from 10 even numbers in multiple of 3, which is \( \binom{10}{3} \). Then, find out combinations of two odd and one even. There are 10 odd and 10 even numbers. So, favourable combinations = \( \binom{10}{2} \times \binom{10}{1} \). Sum up both these combinations.
03

Calculate Probability for Part (a)

Now, calculate the probability for their sum being even by dividing the favourable outcomes by total outcomes.
04

Calculate Favourable Combinations for Part (b)

Next, for the product being even, it is important to note that as long as one number is even, the product will be even, Hence, consider the combinations where all three chosen numbers are odd, which is \( \binom{10}{3} \). Subtract these combinations from the total combinations to get the favourable outcomes for part (b).
05

Calculate Probability for Part (b)

Calculate the probability for their product being even by dividing the favourable outcomes by total outcomes.

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