Find the value of 𝑟r such that the ratio of 3𝑃𝑟3 P r​ and 4𝑃𝑟−14 P r−1​ will be 1221​ ?

1. Break Down the Problem

We need to find the value of $r$ such that the ratio of $3P_r$ and $4P_{r-1}$ is equal to 1221.

2. Relevant Concepts

The formula for permutations is given by:

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

Thus, we can express the permutations as:

• $3P_r = \frac{3!}{(3-r)!}$
• $4P_{r-1} = \frac{4!}{(4 - (r - 1))!} = \frac{4!}{(5 - r)!}$

3. Analysis and Detail

We can set up the equation based on the given ratio:

$\frac{3P_r}{4P_{r-1}} = 1221$

Substituting the permutations into the equation:

$\frac{\frac{3!}{(3-r)!}}{\frac{4!}{(5-r)!}} = 1221$

This simplifies to:

$\frac{3! \cdot (5-r)!}{4! \cdot (3-r)!} = 1221$

Now, substitute the factorial values:

$\frac{6 \cdot (5-r)!}{24 \cdot (3-r)!} = 1221$

Simplifying further gives:

$\frac{(5-r)!}{4 \cdot (3-r)!} = 1221$

By rewriting $(5-r)!$:

$(5-r)(4-r)(3-r)! = 1221 \cdot 4 \cdot (3-r)!$

Canceling $(3-r)!$ results in:

$(5-r)(4-r) = 4884$

4. Verify and Summarize

Now let's expand and rewrite the equation:

$20 - 9r + r^2 = 4884$ $r^2 - 9r + 20 - 4884 = 0$ $r^2 - 9r - 4864 = 0$

Using the quadratic formula, $r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a = 1, b = -9, c = -4864$:

1. Calculate the discriminant: $(-9)^2 - 4 \cdot 1 \cdot (-4864) = 81 + 19456 = 19437$

2. Calculate $r$: $r = \frac{9 \pm \sqrt{19437}}{2}$ First, calculate $\sqrt{19437}$: $\sqrt{19437} \approx 139.4$ Therefore, $r = \frac{9 \pm 139.4}{2}$

   This gives two potential values: $r_1 \approx \frac{148.4}{2} \approx 74.2$ $r_2 \approx \frac{-130.4}{2} \approx -65.2 \quad (\text{not a valid solution})$

Final Answer

Thus, the valid integer solution for $r$ is approximately: $r = 74$