Question

Suppose there are 5 alike dogs, 6 alike monkeys and 7 alike horses. The number of ways of selecting one or more animals from these is

• A. 362
• B. 363
• C. 336
• D. 335
• E. 337

Answer 1

The Correct Option is D

Given: 5 identical dogs, 6 identical monkeys, and 7 identical horses.

We need to find the number of ways to select one or more animals from these groups.

For each type of animal, the number of selection options is:

• Dogs: Can select 0 to 5 (6 options)
• Monkeys: Can select 0 to 6 (7 options)
• Horses: Can select 0 to 7 (8 options)

 

Total number of combinations (including selecting none): \[ 6 \times 7 \times 8 = 336 \]

Since we want at least one animal, we subtract the case where we select none: \[ 336 - 1 = 335 \]

Therefore, the number of ways to select one or more animals is 335.

The correct answer is (D) 335.

Answer 2

Let's denote the number of alike dogs as \( d = 5 \), the number of alike monkeys as \( m = 6 \), and the number of alike horses as \( h = 7 \). We want to select one or more animals.

Since the animals are alike within their species, we can think of this as choosing a number of dogs (0 to 5), a number of monkeys (0 to 6), and a number of horses (0 to 7).

The number of ways to choose dogs is 6 (0 to 5 dogs).

The number of ways to choose monkeys is 7 (0 to 6 monkeys).

The number of ways to choose horses is 8 (0 to 7 horses).

The total number of ways to select animals, including the possibility of selecting zero animals, is the product of the number of choices for each type of animal:

\[ 6 \cdot 7 \cdot 8 = 336 \]

However, we must subtract the case where we select zero animals of each type (i.e., we select no animals at all). This case is only 1 way.

Therefore, the number of ways to select one or more animals is \( 336 - 1 = 335 \).