If `3 * ^(n_1-n_2)P_2=^(n_1+n_2)P_2=90` then the ordered pair `(n_1 ,n_2)` is

Options

A. (8,2)
B. (7,3)
C. (16,8)
D. (9,1)

Answer

To solve the problem, we start with the given equations:

1. \( 3 \cdot P(n_1 - n_2, 2) = 90 \)
2. \( P(n_1 + n_2, 2) = 90 \)

Where \( P(n, r) \) represents the number of permutations of \( n \) items taken \( r \) at a time, given by the formula:
\[ P(n, r) = \frac{n!}{(n - r)!} \]

### Step 1: Rewrite the equations using the permutation formula
From the first equation:
\[ 3 \cdot P(n_1 - n_2, 2) = 90 \]
Using the permutation formula:
\[ P(n_1 - n_2, 2) = \frac{(n_1 - n_2)!}{(n_1 - n_2 - 2)!} = (n_1 - n_2)(n_1 - n_2 - 1) \]
Thus, we can rewrite the first equation as:
\[ 3 \cdot (n_1 - n_2)(n_1 - n_2 - 1) = 90 \]
Dividing both sides by 3:
\[ (n_1 - n_2)(n_1 - n_2 - 1) = 30 \quad \text{(Equation 1)} \]

From the second equation:
\[ P(n_1 + n_2, 2) = 90 \]
Using the permutation formula:
\[ (n_1 + n_2)(n_1 + n_2 - 1) = 90 \quad \text{(Equation 2)} \]

### Step 2: Solve Equation 1
Let \( x = n_1 - n_2 \). Then, we have:
\[ x(x - 1) = 30 \]
This simplifies to:
\[ x^2 - x - 30 = 0 \]
Now, we can factor this quadratic equation:
\[ (x - 6)(x + 5) = 0 \]
Thus, \( x = 6 \) or \( x = -5 \).

Since \( n_1 - n_2 \) must be non-negative, we take:
\[ n_1 - n_2 = 6 \quad \text{(Equation 3)} \]

### Step 3: Solve Equation 2
Now, substituting \( n_1 + n_2 \) into Equation 2:
Let \( y = n_1 + n_2 \). Then we have:
\[ y(y - 1) = 90 \]
This simplifies to:
\[ y^2 - y - 90 = 0 \]
Factoring this quadratic equation:
\[ (y - 10)(y + 9) = 0 \]
Thus, \( y = 10 \) or \( y = -9 \).

Again, since \( n_1 + n_2 \) must be non-negative, we take:
\[ n_1 + n_2 = 10 \quad \text{(Equation 4)} \]

### Step 4: Solve the system of equations
Now we have two equations:
1. \( n_1 - n_2 = 6 \) (Equation 3)
2. \( n_1 + n_2 = 10 \) (Equation 4)

We can solve these equations simultaneously. Adding Equation 3 and Equation 4:
\[ (n_1 - n_2) + (n_1 + n_2) = 6 + 10 \]
This simplifies to:
\[ 2n_1 = 16 \]
Thus:
\[ n_1 = 8 \]

Now substituting \( n_1 = 8 \) back into Equation 4:
\[ 8 + n_2 = 10 \]
So:
\[ n_2 = 2 \]

### Final Answer
The ordered pair \( (n_1, n_2) \) is \( (8, 2) \).