Question

In how many ways at least one horse and at least one dog can be selected out of 8 horses and 7 dogs?
A.${{2}^{15}}-2$
B.${{2}^{15}}-1$
C.$({{2}^{8}}-1)({{2}^{7}}-1)$
D.${}^{15}{{C}_{2}}$

Answer 1

Hint: Total number of ways to select an object from the same n objects is${{2}^{n}}$. Here we have two objects so we will split it into two parts and find numbers of ways to choose the object. Finally we will combine both results to get the required result.

Complete step-by-step answer:
There are 8 horses and 7 dogs in which we need to select at least one horse and one dog.
First find out of ways to select at least one horse from 8 horses.
Total number of ways to select horses from 8 horses is${{2}^{8}}$.
Number of ways to not select any horse is 1.
We need to select at least one horse so the number of ways to select at least one horse is ${{2}^{8}}-1$.
First find out of ways to select at least one dog from 7 dogs.
Total number of ways to select dogs from 7 dogs is${{2}^{7}}$.
Number of ways to not select any dogs is 1.
We need to select at least one dog so the number of ways to select at least one dog is ${{2}^{7}}-1$.
But we need a number of ways to select at least one horse and at least one dogs from 8 horses and 7 dogs together.
Number of ways to select at least one horse and at least one dog from 8 horses and 7 dogs is $({{2}^{8}}-1)({{2}^{7}}-1)$ .
Option (C) is correct.

Note:Here we are not asked for a number of ways to select r objects from n objects, that’s why we will not use nCr formula. There are two different objects to select that’s why for required numbers we will multiply both ways.