QJEEAdv12Maa450: JEE Advanced 2012 Mathematics - Class 12 JEEAdv12Maa450 Solution: Four fair dice and ; each having six faces numbered and are…

For the given condition, the sample space $=6^4$

For favourable condition

Case I : For $D_4$ there are ${ }^6 C_1$ way. Now, it appears on any one of $D_1, D_2, D_3$ i.e. ${ }^3 C_1 \cdot 1$ , for other two there are $5 \times 5$ ways.

$\Rightarrow{ }^6 \mathrm{C}_1 \cdot{ }^3 \mathrm{C}_1 \cdot 1 \cdot 5^2=450 \text { Ways }$

Case II : For $\mathrm{D}_4$ there are ${ }^6 \mathrm{C}_1$ ways. Now, it appears on any two of $D_1, D_2, D_3$ i.e. ${ }^3 \mathrm{C}_2 .1 .1$ , for other one there are 5 ways.

$\Rightarrow{ }^6 \mathrm{C}_1 \cdot{ }^3 \mathrm{C}_2 \cdot 1^2 \cdot 5=90 \text { Ways }$

Case III : For $\mathrm{D}_4$ there are ${ }^6 \mathrm{C}_1$ ways. Now, it appears on all i.e. $1^3$ .

$\Rightarrow{ }^6 \mathrm{C}_1 \cdot 1^3=6$

Total number of favourable cases

$\begin{aligned}& =450+90+6 \\& =546 \text { ways }\end{aligned}$

Thus, Probability $=\frac{546}{6^4}=\frac{91}{216}$