A box contains 10 tiles labeled with the integers from 1 through 10. If 2 tiles are drawn … — ACT

A box contains $10$ tiles labeled with the integers from $1$ through $10$ . If $2$ tiles are drawn at random without replacement, how many distinct pairs of tiles (where the order of selection does not matter) have a sum that is an odd number?

Answer: 25 pairs

Answer

The total number of distinct pairs with an odd sum is 25.

To obtain an odd sum when adding two integers, one integer must be odd and the other must be even. In the range

1

through

10

, there are

5

odd integers (

1, 3, 5, 7, 9

) and

5

even integers (

2, 4, 6, 8, 10

). To form a pair with an odd sum, one tile must be selected from the

5

odd tiles and one tile must be selected from the

5

even tiles. By the Fundamental Counting Principle, the number of such distinct pairs is

5 \times 5 = 25

Step-by-Step Solution

Determine the parity condition for an odd sum

One tile must be odd and the other must be even

The sum of two integers is odd if and only if one addend is odd and the other addend is even.

Count the number of odd and even options

5 odd tiles and 5 even tiles

Among the integers

1

through

10

, the odd numbers are

1, 3, 5, 7, 9

(

5

total) and the even numbers are

2, 4, 6, 8, 10

(

5

total).

Apply the Fundamental Counting Principle

5 × 5 = 25 distinct pairs

Selecting one odd tile out of 5 possibilities and one even tile out of 5 possibilities gives

5 \times 5 = 25

distinct unordered pairs.

Key Concept

Fundamental Counting Principle and Parity of Integers

Estimated Time:1m 0s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic