Answer these questions based on the following informations
After completing his MBA, Zafar shifted to a new flat where he bought a new bookself and a set of 7 novels. The names of these novels, In the chronoligical order, are A,B,C,D,E, F and G
If he wants to keep at least 4 novels in his bookshelf, find the number of ways in which he can arrange the novels

Options

A. 13440
B. 1440
C. 12440
D. 840

Answer

To solve the problem of how many ways Zafar can arrange at least 4 novels from his collection of 7 novels (A, B, C, D, E, F, G), we can break it down into steps:

### Step 1: Understand the problem
Zafar has 7 novels and wants to arrange at least 4 of them on his bookshelf. This means he can choose to arrange 4, 5, 6, or all 7 novels.

### Step 2: Use the formula for permutations
The number of ways to arrange \( r \) novels from \( n \) novels is given by the formula:
\[ nPr = \frac{n!}{(n-r)!} \]
where \( n! \) (n factorial) is the product of all positive integers up to \( n \).

### Step 3: Calculate the arrangements for each case
1. **Arranging 4 novels**:
\[ 7P4 = \frac{7!}{(7-4)!} = \frac{7!}{3!} = \frac{7 \times 6 \times 5 \times 4 \times 3!}{3!} = 7 \times 6 \times 5 \times 4 = 840 \]

2. **Arranging 5 novels**:
\[ 7P5 = \frac{7!}{(7-5)!} = \frac{7!}{2!} = \frac{7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{2 \times 1} = 7 \times 6 \times 5 \times 4 \times 3 = 2520 \]

3. **Arranging 6 novels**:
\[ 7P6 = \frac{7!}{(7-6)!} = \frac{7!}{1!} = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040 \]

4. **Arranging all 7 novels**:
\[ 7P7 = \frac{7!}{(7-7)!} = \frac{7!}{0!} = 7! = 5040 \]

### Step 4: Sum all the arrangements
Now, we add all the arrangements together:
\[ Total = 7P4 + 7P5 + 7P6 + 7P7 = 840 + 2520 + 5040 + 5040 \]
Calculating this gives:
\[ Total = 840 + 2520 + 5040 + 5040 = 12960 \]

### Final Answer
Thus, the total number of ways Zafar can arrange at least 4 novels is **12960**.

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