There are N beads in a bag, of which 6 are red. Two beads are selected at random from the bag. The probability that both beads selected are red is 0.125. 

Show that N^2-N-240=0 

Hence find the value of N 

We are given that there are \( N \) beads in a bag, out of which 6 are red. The probability of selecting two red beads at random is \( 0.125 \).

### Step 1: Expression for the probability
The probability of selecting two red beads can be computed using combinations. The number of ways to select 2 red beads from 6 red beads is:

\[
\binom{6}{2} = \frac{6 \times 5}{2} = 15
\]

The total number of ways to select 2 beads from \( N \) beads is:

\[
\binom{N}{2} = \frac{N \times (N-1)}{2}
\]

The probability of selecting two red beads is the ratio of these two combinations:

\[
P(\text{both red}) = \frac{\binom{6}{2}}{\binom{N}{2}} = \frac{15}{\frac{N \times (N-1)}{2}} = \frac{30}{N(N-1)}
\]

We are told that this probability is \( 0.125 \):

\[
\frac{30}{N(N-1)} = 0.125
\]

### Step 2: Solve the equation

Now, multiply both sides of the equation by \( N(N-1) \):

\[
30 = 0.125 \times N(N-1)
\]

Simplifying \( 0.125 \) as \( \frac{1}{8} \), we get:

\[
30 = \frac{1}{8} \times N(N-1)
\]

Multiplying both sides by 8 to eliminate the fraction:

\[
240 = N(N-1)
\]

### Step 3: Rearranging into quadratic form

Expand the right-hand side:

\[
240 = N^2 - N
\]

Rearrange the equation:

\[
N^2 - N - 240 = 0
\]

This is the required quadratic equation.

### Step 4: Solve the quadratic equation

Now, solve \( N^2 - N - 240 = 0 \) using the quadratic formula:

\[
N = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(1)(-240)}}{2(1)}
\]

Simplifying:

\[
N = \frac{1 \pm \sqrt{1 + 960}}{2} = \frac{1 \pm \sqrt{961}}{2} = \frac{1 \pm 31}{2}
\]

Thus, \( N \) can be:

\[
N = \frac{1 + 31}{2} = 16 \quad \text{or} \quad N = \frac{1 - 31}{2} = -15
\]

Since \( N \) must be positive, we have:

\[
N = 16
\]

### Final Answer:
The value of \( N \) is 16.

Do you want more details or have any questions? Here are five related questions for further exploration:

1. What would the probability be if three beads were selected instead of two?
2. If there were 8 red beads, how would that change the equation and solution for \( N \)?
3. How can this method be applied if there are two different colors of beads in the bag?
4. What happens to the probability if we only replace one bead after each selection?
5. How would this probability change if the beads were selected one by one instead of simultaneously?

**Tip**: When solving quadratic equations, always check the discriminant (\(b^2 - 4ac\)) to ensure the equation has real solutions.

There are N beads in a bag, of which 6 are red. Two beads are selected at random from the bag. The probability that both beads selected are red is 0.125. Show that N^2 - N - 240 = 0. Hence find the value of N.

Learn how to solve a quadratic equation, N^2 - N - 240 = 0, arising from a probability problem involving selecting two red beads from a bag. This detailed step-by-step solution explains combinatorics and the quadratic formula, suitable for students in Grades 9-11.

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