Question

If p and q are positive integers such that \(^{p+q}P_2=42\ and\ ^{p-q}P_2=20\), then the values of p and q respectively

• A. 6,1
• B. 4,3
• C. 7,2
• D. 5,2
• E. 7,5

Answer 1

The Correct Option is D

We are given that: 

• \(^{p+q}P_2 = 42\)
• \(^{p-q}P_2 = 20\)

The formula for \(^nP_r\) is:

\(^nP_r = \frac{n!}{(n-r)!}\)

We start with the first equation: \(^{p+q}P_2 = 42\).

Using the formula:

\(^{p+q}P_2 = \frac{(p+q)!}{(p+q-2)!}\)

We are given that this equals 42, so:

\(\frac{(p+q)(p+q-1)}{2} = 42\)

Multiplying both sides by 2:

\((p+q)(p+q-1) = 84\)

We now proceed with the second equation: \(^{p-q}P_2 = 20\).

\(^{p-q}P_2 = \frac{(p-q)(p-q-1)}{2}\)

We are given that this equals 20, so:

\(\frac{(p-q)(p-q-1)}{2} = 20\)

Multiplying both sides by 2:

\((p-q)(p-q-1) = 40\)

Now we have the following system of equations:

• \((p+q)(p+q-1) = 84\)
• \((p-q)(p-q-1) = 40\)

We solve these equations by trial and error or algebraically.

If we try \(p+q = 7\), we get:

\((7)(6) = 42\), which is correct.

For \(p-q = 5\), we get:

\((5)(4) = 20\), which is correct.

Therefore, \(p = 5\) and \(q = 2\).

The values of \(p\) and \(q\) are 5 and 2, respectively.