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Question: There are 25 buttons in another bag. This bag contains \[x\] blue button. Two buttons are taken at r...

There are 25 buttons in another bag. This bag contains xx blue button. Two buttons are taken at random without replacement. The probability that they are both blue is 7100.\dfrac{7}{{100}}. Show that x2−x−42=0{x^2} - x - 42 = 0

Explanation

Solution

At first, we will find the probability to draw the blue button among the total blue buttons and the total number of buttons.
Then, we will equate with the given probability.
Finally, we can prove.

Complete step-by-step solution:
It is given that; there are 2525 buttons in another bag. This bag contains xx blue button. Two buttons are taken at random without replacement. The probability that they are both blue is 7100.\dfrac{7}{{100}}.
We have to show that, x2−x−42=0{x^2} - x - 42 = 0.
Since, the number of total buttons is 2525 and the number of blue buttons is xx.
Two blue buttons will be drawn at random.
So, the number of ways to draw a blue button among xx is xC1^x{C_1}.
After drawing a button, the number of blue buttons is x−1x - 1.
So, So, the number of ways to draw a blue button among x−1x - 1 blue buttons are x−1C1^{x - 1}{C_1}.
Now, the number of ways to draw a blue button among the total number of buttons 2525 is 25C1^{25}{C_1}.
After drawing a button, the number of buttons is 25−1=2425 - 1 = 24.
So, the number of ways to draw a blue button among 2424 buttons are 24C1^{24}{C_1}.
So, the required probability is xC1.x−1C125C1.24C1=7100\dfrac{{^x{C_1}{.^{x - 1}}{C_1}}}{{^{25}{C_1}{.^{24}}{C_1}}} = \dfrac{7}{{100}}
Simplifying we get,
\Rightarrow$$$\dfrac{{x(x - 1)}}{{25 \times 24}} = \dfrac{7}{{100}}.$$ Simplifying again we get, \Rightarrow\dfrac{{x(x - 1)}}{{1 \times 24}} = \dfrac{7}{{4}}$$ Simplifying again we get, $\Rightarrow{x^2} - x = 42 Hence, $\Rightarrow$$${x^2} - x - 42 = 0
Hence, proved.

Note: We already know that, probability means possibility. It is a branch of mathematics that deals with the occurrence of a random event.
The value is expressed from zero to one. Probability has been introduced in Maths to predict how likely events are to happen.
The probability formula is defined as the possibility of an event to happen is equal to the ratio of the number of favourable outcomes and the total number of outcomes.