A bouquet has to be formed from 18 different flowers so that it should contain not less than three flowers. How many ways are there of doing this in?

Options

A. 5,24,288
B. 2,62,144
C. 2,61,972
D. None of these

Answer

To solve the problem of forming a bouquet from 18 different flowers with the condition that it should contain at least 3 flowers, we can follow these steps:

### Step 1: Understand the Requirement
We need to form a bouquet using at least 3 flowers from a total of 18 different flowers. This means we can choose 3, 4, 5, up to 18 flowers.

### Step 2: Use Combinations
Since the order of flowers in the bouquet does not matter, we will use combinations. The number of ways to choose r flowers from n flowers is given by the combination formula:

\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]

### Step 3: Calculate the Total Combinations
We need to calculate the combinations for choosing 3 to 18 flowers:

\[ \text{Total Ways} = \binom{18}{3} + \binom{18}{4} + \binom{18}{5} + \ldots + \binom{18}{18} \]

### Step 4: Calculate Each Combination
Now we will calculate each combination:

1. \(\binom{18}{3} = \frac{18!}{3!(18-3)!} = \frac{18 \times 17 \times 16}{3 \times 2 \times 1} = 816\)
2. \(\binom{18}{4} = \frac{18!}{4!(18-4)!} = \frac{18 \times 17 \times 16 \times 15}{4 \times 3 \times 2 \times 1} = 3060\)
3. \(\binom{18}{5} = \frac{18!}{5!(18-5)!} = \frac{18 \times 17 \times 16 \times 15 \times 14}{5 \times 4 \times 3 \times 2 \times 1} = 8568\)
4. \(\binom{18}{6} = 18564\)
5. \(\binom{18}{7} = 31824\)
6. \(\binom{18}{8} = 43758\)
7. \(\binom{18}{9} = 48620\)
8. \(\binom{18}{10} = 43758\)
9. \(\binom{18}{11} = 31824\)
10. \(\binom{18}{12} = 18564\)
11. \(\binom{18}{13} = 8568\)
12. \(\binom{18}{14} = 3060\)
13. \(\binom{18}{15} = 816\)
14. \(\binom{18}{16} = 153\)
15. \(\binom{18}{17} = 18\)
16. \(\binom{18}{18} = 1\)

### Step 5: Sum All Combinations
Now we will sum all these values:

\[ \text{Total Ways} = 816 + 3060 + 8568 + 18564 + 31824 + 43758 + 48620 + 43758 + 31824 + 18564 + 8568 + 3060 + 816 + 153 + 18 + 1 \]

Calculating this gives us:

\[ \text{Total Ways} = 261972 \]

### Final Answer
Thus, the total number of ways to form the bouquet with at least 3 flowers from 18 different flowers is **261972**.

---