NIMCET 2023 — Mathematics PYQ

To find the probability that no black ball is chosen when drawing 3 balls, we need to determine the number of favorable outcomes and divide it by the total number of possible outcomes.

Step 1: Calculate the total number of balls in the bag

• Number of yellow balls = $5$

• Number of black balls = $4$

• Number of green balls = $3$

$$\text{Total number of balls}=5+4+3=12$$

Step 2: Find the total number of ways to draw 3 balls out of 12

The total number of ways to select 3 balls from 12 is given by the combination formula $\left(\genfrac{}{}{0ex}{}{n}{r}\right)={}^n{C}_r$:

$$\text{Total Outcomes }[n(S)]={}^{12}{C}_3$$

$$n(S)=\frac{12\times 11\times 10}{3\times 2\times 1}=2\times 11\times 10=220$$

Step 3: Find the number of ways to choose 3 balls such that NO black ball is selected

If no black ball is to be chosen, all 3 balls must be selected from the remaining yellow and green balls.

$$\text{Total non-black balls}=\text{Yellow}+\text{Green}=5+3=8$$

The number of favorable ways to choose 3 balls from these 8 non-black balls is:

$$\text{Favorable Outcomes }[n(E)]={}^8{C}_3$$

$$n(E)=\frac{8\times 7\times 6}{3\times 2\times 1}=8\times 7=56$$

Step 4: Calculate the probability

The probability $P(E)$ of an event is given by the formula:

$$P(E)=\frac{n(E)}{n(S)}$$

$$P(E)=\frac{56}{220}$$

Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 4:

$$P(E)=\frac{56÷4}{220÷4}=\frac{14}{55}$$

Correct Answer: A) $14/55$