Consider the sequence `('^(n)C_(0))/(1.2.3),('^(n)C_(1))/(2.3.4),('^(n)C_(2))/(3.4.5),....,` if `n=50` then greatest term is

Options

A. `30^(th)`
B. `24^(th)`
C. `26^(th)`
D. `27^(th)`

Explanation

To solve the problem, we need to analyze the sequence given and determine which term is the greatest when \( n = 50 \). The sequence is defined as:

\[ T_r = \frac{nC_r}{(r+1)(r+2)(r+3)} \]

where \( n = 50 \).

### Step 1: Define the General Term
The general term of the sequence can be expressed as:

\[ T_{r+1} = \frac{50C_r}{(r+1)(r+2)(r+3)} \]

### Step 2: Write the Binomial Coefficient
The binomial coefficient \( 50C_r \) can be expressed as:

\[ 50C_r = \frac{50!}{r!(50-r)!} \]

### Step 3: Substitute the Binomial Coefficient into the General Term
Substituting the expression for \( 50C_r \) into the general term, we get:

\[ T_{r+1} = \frac{50!}{r!(50-r)! \cdot (r+1)(r+2)(r+3)} \]

### Step 4: Simplify the General Term
We can rewrite the denominator:

\[ (r+1)(r+2)(r+3) = (r+3)! \]

Thus, we can express \( T_{r+1} \) as:

\[ T_{r+1} = \frac{50!}{(50-r)! \cdot (r+3)!} \]

### Step 5: Identify the Maximum Term
To find the maximum term, we need to analyze the behavior of \( T_{r+1} \). The term \( T_{r+1} \) will be maximized when \( r \) is approximately \( \frac{n}{2} \).

### Step 6: Set Up the Condition for Maximum
Since \( n = 50 \):

\[ r + 3 = \frac{53}{2} \]

This gives:

\[ r + 3 = 26.5 \implies r = 23.5 \]

Since \( r \) must be an integer, we can take \( r = 23 \).

### Step 7: Find the Corresponding Term
The term corresponding to \( r = 23 \) is:

\[ T_{24} = \frac{50C_{23}}{(24)(25)(26)} \]

### Step 8: Conclusion
Thus, the greatest term in the sequence when \( n = 50 \) is the 24th term.

### Final Answer
The greatest term is the **24th term**.

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