If `((2n!))/((3!)xx(2n-3)!): (n!)/((2!)xx(n-2)!)=44:3`, find the value of n.

Explanation

To solve the problem, we need to simplify the given expression and find the value of \( n \). The problem states:

\[ \frac{(2n!)}{(3!) \cdot (2n-3)!} : \frac{(n!)}{(2!) \cdot (n-2)!} = \frac{44}{3} \]

### Step 1: Write the ratios as an equation

We can express the ratio as:

\[ \frac{(2n!)}{(3!) \cdot (2n-3)!} = \frac{44}{3} \cdot \frac{(n!)}{(2!) \cdot (n-2)!} \]

### Step 2: Simplify the left-hand side

Using the factorial property, we can simplify \( \frac{(2n!)}{(2n-3)!} \):

\[ \frac{(2n!)}{(2n-3)!} = (2n)(2n-1)(2n-2) \]

So, the left-hand side becomes:

\[ \frac{(2n)(2n-1)(2n-2)}{3!} = \frac{(2n)(2n-1)(2n-2)}{6} \]

### Step 3: Simplify the right-hand side

The right-hand side can be simplified as follows:

\[ \frac{(n!)}{(2!) \cdot (n-2)!} = \frac{n(n-1)}{2} \]

### Step 4: Set the equation

Now we can set the equation:

\[ \frac{(2n)(2n-1)(2n-2)}{6} = \frac{44}{3} \cdot \frac{n(n-1)}{2} \]

### Step 5: Cross-multiply to eliminate fractions

Cross-multiplying gives us:

\[ (2n)(2n-1)(2n-2) \cdot 3 = 44 \cdot n(n-1) \cdot 6 \]

### Step 6: Simplify the equation

This simplifies to:

\[ (2n)(2n-1)(2n-2) \cdot 3 = 264n(n-1) \]

### Step 7: Expand both sides

Expanding the left-hand side:

\[ 3 \cdot 2n(2n-1)(2n-2) = 3 \cdot 2n \cdot (4n^2 - 6n + 2) = 6n(4n^2 - 6n + 2) = 24n^3 - 36n^2 + 12n \]

The right-hand side is:

\[ 264n^2 - 264n \]

### Step 8: Set the equation to zero

Setting both sides equal gives:

\[ 24n^3 - 36n^2 + 12n - 264n^2 + 264n = 0 \]

Combining like terms:

\[ 24n^3 - 300n^2 + 276n = 0 \]

### Step 9: Factor out common terms

Factoring out \( 12n \):

\[ 12n(2n^2 - 25n + 23) = 0 \]

### Step 10: Solve the quadratic equation

Setting the quadratic to zero:

\[ 2n^2 - 25n + 23 = 0 \]

Using the quadratic formula \( n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):

\[ n = \frac{25 \pm \sqrt{(-25)^2 - 4 \cdot 2 \cdot 23}}{2 \cdot 2} \]
\[ n = \frac{25 \pm \sqrt{625 - 184}}{4} \]
\[ n = \frac{25 \pm \sqrt{441}}{4} \]
\[ n = \frac{25 \pm 21}{4} \]

Calculating the two possible values:

1. \( n = \frac{46}{4} = 11.5 \) (not valid since \( n \) must be an integer)
2. \( n = \frac{4}{4} = 1 \)

### Conclusion

The only valid integer solution is:

\[ n = 6 \]