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 start by selecting 8 objects from the total of 21 objects. The number of ways to choose 8 objects from 21 is given by the combination formula:

\[ \binom{21}{8} \]

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

\[ \binom{13}{7} \]

### Step 3: The last group of 6 objects
After selecting the second group of 7 objects, we will have 6 objects left (13 - 7 = 6). These remaining 6 objects will automatically form the last group. The number of ways to choose all 6 objects from 6 is:

\[ \binom{6}{6} = 1 \]

### Step 4: Calculate the total number of arrangements
Now, we can find the total number of ways to group the objects by multiplying the number of ways to choose each group:

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

### Step 5: Simplifying the expression
We know that:

\[ \binom{6}{6} = 1 \]

Thus, we can simplify our expression to:

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

### Step 6: Using the factorial formula
The combination formula is given by:

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

Using this, we can express our combinations as:

\[ \binom{21}{8} = \frac{21!}{8! \cdot (21-8)!} = \frac{21!}{8! \cdot 13!} \]

\[ \binom{13}{7} = \frac{13!}{7! \cdot (13-7)!} = \frac{13!}{7! \cdot 6!} \]

### Step 7: Plugging back into the total ways
Now substituting these back into our total ways expression gives:

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

The \(13!\) cancels out:

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

### Final Answer
Thus, the number of ways to group 21 objects into three groups of 8, 7, and 6 is:

\[ \frac{21!}{8! \cdot 7! \cdot 6!} \]