# CRE 8 - Selecting books from a shelf | Modern Math - Permutation & Combination

**Answer the next 2 questions based on the information given below:**

On a library shelf, there are 5 different books for Maths, 4 different books for Reasoning & 3 different books for English. In how many ways can a person select

**CRE 8 - Selecting books from a shelf | Modern Math - Permutation & Combination**

A book from the shelf?

Answer: 12

**Explanation** :

There is a total of 5 + 4 + 3 = 12 different books on the shelf.

We have to pick one book.

∴ Number of ways of picking a book = ^{12}C_{1} = 12 ways.

Hence, 12.

Workspace:

**CRE 8 - Selecting books from a shelf | Modern Math - Permutation & Combination**

At least one book from the shelf?

Answer: 4095

**Explanation** :

There is a total of 5 + 4 + 3 = 12 different books on the shelf.

We have to pick at least book. Let us name these books as

Maths: M_{1}, M_{2}, M_{3}, M_{4} and M_{5}.

Reasoning: R_{1}, R_{2}, R_{3} and R_{4}.

English: E_{1}, E_{2} and E_{3}.

There are two possibilities for each of these books i.e., either the book will be picked or not picked.

∴ Total possibilities for picking books = 2^{12}.

Now out of these 2^{12} possibilities, there would be one possibility when no book is picked.

∴ Number of ways of picking at least 1 book = 2^{12} - 1 = 4096 – 1 = 4095 ways.

Hence, 4095.

Workspace:

**CRE 8 - Selecting books from a shelf | Modern Math - Permutation & Combination**

At least one book on each subject from the shelf?

Answer: 3255

**Explanation** :

There is a total of 5 + 4 + 3 = 12 different books on the shelf.

We have to pick at least book. Let us name these books as

Maths: M_{1}, M_{2}, M_{3}, M_{4} and M_{5}.

Reasoning: R_{1}, R_{2}, R_{3} and R_{4}.

English: E_{1}, E_{2} and E_{3}.

There are two possibilities for each of the Maths i.e., either the book will be picked or not picked.

∴ Total possibilities for picking Maths books = 2^{5}.

Now out of these 25 possibilities, there would be one possibility when no book is picked.

∴ Number of ways of picking at least 1 Maths book = 2^{5} – 1.

Similarly,

Number of ways of picking at least 1 Reasoning book = 2^{4} – 1, and

Number of ways of picking at least 1 English book = 2^{3} – 1.

∴ Number of ways of picking at least 1 book of each subject = (2^{5} – 1) × (2^{4} – 1) × (2^{3} – 1) = 31 × 15 × 7 = 3255.

Hence, 3255.

Workspace:

**Answer the next 2 questions based on the information given below:**

On a library shelf, there are 5 identical books for Maths, 4 identical books for Reasoning & 3 identical books for English. In how many ways can a person select

**CRE 8 - Selecting books from a shelf | Modern Math - Permutation & Combination**

A book from the shelf?

Answer: 3

**Explanation** :

There are 5 identical Maths books, 4 identical Reasoning books and 3 identical English books.

We have to pick 1 book, now this book can either be a Math book or Reasoning book or an English book i.e., 3 ways.

Hence, 3.

Workspace:

**CRE 8 - Selecting books from a shelf | Modern Math - Permutation & Combination**

At least one book from the shelf?

Answer: 119

**Explanation** :

There are 5 identical Maths books, 4 identical Reasoning books and 3 identical English books.

For Math books:

Either we can pick no Math book or only 1 book or 2 books or 3 books or 4 books or 5 books i.e., we can pick Math books in (5 + 1) = 6 ways.

Similarly for Reasoning, number of ways of picking books = (4 + 1) = 5 ways, and

For English, number of ways of picking books = (3 + 1) = 4 ways.

∴ Total Number of ways of picking books = (5 + 1)(4 + 1)(3 + 1) = 120 ways.

Out of these 120 ways, there would be 1 possibility when no book is picked.

∴ Total Number of ways of picking at least 1 book = 120 – 1 = 119 ways.

Hence, 119.

Workspace:

**CRE 8 - Selecting books from a shelf | Modern Math - Permutation & Combination**

At least one book on each subject from the shelf?

Answer: 60

**Explanation** :

There are 5 identical Maths books, 4 identical Reasoning books and 3 identical English books.

For Math books:

Either we can pick only 1 book or 2 books or 3 books or 4 books or 5 books i.e., we can pick Math books in 5 ways.

Similarly for Reasoning, number of ways of picking books = 4 ways, and

For English, number of ways of picking books = 3 ways.

∴ Total Number of ways of picking at least one book of each subject = 5 × 4 × 3 = 60 ways.

Hence, 60.

Workspace:

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