Answer these questions based on the following information.
Last Diwali Chhota Bheem got 10 distinct coloured candles placed in a row. Then he asked his friend to lit 3 candles.
Find the number of ways of selection of candles so that all the 3 candles are not consecutive.

Options

A. 112
B. 12
C. 132
D. 66

Answer

To find the number of ways to select 3 candles from 10 distinct colored candles such that no two selected candles are consecutive, we can follow these steps:

### Step 1: Calculate the total number of ways to select 3 candles from 10.
The total number of ways to choose 3 candles from 10 is given by the combination formula:

\[ \text{Total Ways} = \binom{10}{3} = \frac{10!}{3!(10-3)!} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120 \]

### Step 2: Calculate the number of ways to select 3 consecutive candles.
If we consider 3 consecutive candles as a single unit, we can represent this unit as a single "block." Thus, if we have 3 candles in a block, we can visualize it as follows:

- The block of 3 candles takes up 3 positions.
- This leaves us with 7 remaining positions (10 total - 3 in the block = 7).

Now, we can place this block in the row of 10 candles. The block can start at any of the first 8 positions (1 through 8), since if it starts at position 8, it occupies positions 8, 9, and 10.

Thus, the number of ways to select 3 consecutive candles is:

\[ \text{Ways for 3 consecutive candles} = 8 \]

### Step 3: Subtract the number of ways to select 3 consecutive candles from the total ways.
To find the number of ways to select 3 candles such that no two are consecutive, we subtract the number of ways to select 3 consecutive candles from the total number of ways:

\[ \text{Ways with no consecutive candles} = \text{Total Ways} - \text{Ways for 3 consecutive candles} = 120 - 8 = 112 \]

### Final Answer:
The number of ways to select 3 candles such that no two are consecutive is **112**.

---