The number of ways in which 21 objects can be grouped into three groups of 8, 7, and 6 objects,is

Options

A. `(20!)/(8!+7!+6!)`
B. `(21!)/(8!7!)`
C. `(21!)/(8!7!6!)`
D. `(21!)/(8!+7!+6!)`

Answer

To solve the problem of grouping 21 objects into three groups of 8, 7, and 6 objects, we can follow these steps:

### Step 1: Choose the first group of 8 objects
We need to select 8 objects from the total of 21 objects. The number of ways to choose 8 objects from 21 can be calculated using the combination formula:

\[ \text{Number of ways to choose 8 from 21} = \binom{21}{8} = \frac{21!}{8!(21-8)!} = \frac{21!}{8! \cdot 13!} \]

### Step 2: Choose the second group of 7 objects
After selecting the first group of 8 objects, we have 13 objects left (21 - 8 = 13). Now, we need to choose 7 objects from these remaining 13 objects:

\[ \text{Number of ways to choose 7 from 13} = \binom{13}{7} = \frac{13!}{7!(13-7)!} = \frac{13!}{7! \cdot 6!} \]

### Step 3: The last group of 6 objects
After selecting the first two groups, there will be 6 objects remaining (21 - 8 - 7 = 6). Since all remaining objects will form the last group, there is only 1 way to choose this group:

\[ \text{Number of ways to choose 6 from 6} = \binom{6}{6} = 1 \]

### Step 4: Calculate the total number of ways
Now, we multiply the number of ways to choose each group together to find the total number of ways to group the objects:

\[ \text{Total ways} = \binom{21}{8} \times \binom{13}{7} \times \binom{6}{6} \]

Substituting the values we calculated:

\[ \text{Total ways} = \frac{21!}{8! \cdot 13!} \times \frac{13!}{7! \cdot 6!} \times 1 \]

Notice that \(13!\) cancels out:

\[ \text{Total ways} = \frac{21!}{8! \cdot 7! \cdot 6!} \]

### Step 5: Final Calculation
Now we can compute the final value:

\[ \text{Total ways} = \frac{21!}{8! \cdot 7! \cdot 6!} \]