MHTCET 13 Apr 2026 Shift 1 Paper

Concept:The binomial probability $P(X=r)$ is maximum when the binomial coefficient ${}^{6}C_{r}$ is maximum, because the factor $\left(\frac{1}{2}\right)^{6}$ is constant for all values of $r$.Explanation:Given $X \sim B\left(6,\frac{1}{2}\right)$, we have $n=6$, $p=\frac{1}{2}$ and $q=\frac{1}{2}$.The probability mass function is:$$P(X=r)={}^{6}C_{r}\left(\frac{1}{2}\right)^{r}\left(\frac{1}{2}\right)^{6-r}={}^{6}C_{r}\left(\frac{1}{2}\right)^{6}$$Since $\left(\frac{1}{2}\right)^{6}$ is fixed, we only need to find the value of $r$ for which ${}^{6}C_{r}$ is largest.The binomial coefficients are:$${}^{6}C_{0}=1,\; {}^{6}C_{1}=6,\; {}^{6}C_{2}=15,\; {}^{6}C_{3}=20,\; {}^{6}C_{4}=15,\; {}^{6}C_{5}=6,\; {}^{6}C_{6}=1$$The maximum value is $20$, which occurs at $r=3$.Therefore, $P(X=r)$ is maximum at $X=3$.Answer:$X=3$Hence, the correct option is Option C.