IGDTUW 2026 — Mathematics PYQ

This problem can be solved using the Binomial Probability Distribution formula.

The probability of getting exactly $r$ successes in $n$ independent trials is given by:

$$P(X=r){=}^n{C}_r\cdot p^r\cdot q^{n-r}$$

Where:

• $n=$ total number of trials (number of coins tossed) $=8$

• $r=$ number of successful outcomes (number of heads required) $=6$

• $p=$ probability of getting a head in a single toss $=\frac{1}{2}$

• $q=$ probability of getting a tail in a single toss $=1-p=\frac{1}{2}$

Step 1: Substitute the values into the formula

$$P(X=6){=}^8{C}_6\cdot {\left(\frac{1}{2}\right)}^6\cdot {\left(\frac{1}{2}\right)}^{8-6}$$

$$P(X=6){=}^8{C}_6\cdot {\left(\frac{1}{2}\right)}^6\cdot {\left(\frac{1}{2}\right)}^2$$

$$P(X=6){=}^8{C}_6\cdot {\left(\frac{1}{2}\right)}^8$$

Step 2: Calculate the combinations ${}^8{C}_6$

Using the combination formula ${}^n{C}_r=\frac{n!}{r!(n-r)!}$:

$${}^8{C}_6=\frac{8!}{6!\cdot 2!}=\frac{8\cdot 7}{2\cdot 1}=28$$

Step 3: Simplify the expression

Now substitute ${}^8{C}_6=28$ back into the probability equation:

$$P(X=6)=28\cdot \frac{1}{256}$$

$$P(X=6)=\frac{28}{256}$$

Step 4: Reduce to the simplest form

Divide both the numerator and the denominator by 4:

$$P(X=6)=\frac{7}{64}$$

Correct Answer:

(a) $\frac{7}{64}$