Question

A zoo has 20 Zebras, 12 Giraffes, 11 Lions, and 3 Tigers. The number of ways in which a man can visit the zoo so that he must see at least one tiger is
$\begin{aligned} & \left[ a \right]\ 21\times 13\times 12\times 3 \\\ & \left[ b \right]\ 7\times {{2}^{43}} \\\ & \left[ c \right]\ 7\times 21\times 13\times 12-1 \\\ & \left[ d \right]\ 6\times {{2}^{43}} \\\ \end{aligned}$

Answer 1

Hint: Use the fundamental principle of counting, which states that if a task A can be done in m ways and another task B can be done in n ways then the number of ways of doing both the tasks is m+n and the number of ways of doing either of the tasks is m+n. Use the fact that since a person can decide to visit an animal or not to visit an animal, the number of ways in which an animal can be selected by a person is 2. Hence determine the total number of ways of visiting the zoo without any restriction. Also, determine the number of ways in which no tiger has been visited. Hence determine the number of ways in which at least one tiger can be visited.

Complete step-by-step answer:
The number of animals in the zoo is 20+12 +11+3 = 46.
Of these 46 animals, three are tigers.
Now for each animal, the man can decide to visit the animal or not to visit the animal.
Hence for each animal, we have two choices.
Hence the total number of ways in which the animals can be visited is $2\,\times 2\times \cdots \times 2\ \text{ }\left( 46\ \text{times} \right)={{2}^{46}}$
Now, among these possible ways, there are some ways in which no tiger has been visited. We need to subtract those ways. Since in those possible ways, each tiger has only 1 choice(Not Visited), we have the total number of ways in which no tiger has been visited is $2\times 2\times \cdots \times 2\text{ }\left( 43\text{ times} \right)\times 1\times 1\times 1={{2}^{43}}$
Hence the total number of ways in which at least one tiger has been visited is ${{2}^{46}}-{{2}^{43}}={{2}^{43}}\left( {{2}^{3}}-1 \right)=7\times {{2}^{43}}$
Hence option [b] is correct.

Note: Alternative solution-
The number of ways in which we can select 1 tiger among 3 tigers is $=\left( ^{3}{{C}_{1}} \right)$
The number of ways in which we can select 2 tigers among 3 tigers is $=\left( ^{3}{{C}_{2}} \right)$
The number of ways in which we can select 3 tigers among 3 tigers is $=\left( ^{3}{{C}_{3}} \right)$
Hence the number of ways in which at least one tiger among three tigers can be selected $=\left( ^{3}{{C}_{1}}{{+}^{3}}{{C}_{2}}{{+}^{3}}{{C}_{3}} \right)=1+3+1=7$.
Now, of the remaining 43 animals, we can select 0 or 1 or 2 or … or 43 animals.
Hence the number of ways of selecting the remaining animals ${{=}^{43}}{{C}_{0}}{{+}^{43}}{{C}_{1}}+\cdots {{+}^{43}}{{C}_{43}}$
We know that $^{n}{{C}_{0}}{{+}^{n}}{{C}_{1}}+\cdots {{+}^{n}}{{C}_{n}}={{2}^{n}}$
Hence the number of ways of selecting the remaining animals $={{2}^{43}}$
Hence by the fundamental principle of counting, the number of ways in which the man can visit the zoo $={{2}^{43}}\times 7$, which is the same as obtained above
Hence option [b] is correct.