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Two numbers are chosen at random from $\{1,2,3,4,5,6,7,8\}$ at a time. The probability that smaller of the two numbers is less than 4 is
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The correct answer is:
$\frac{9}{14}$
Case I When smaller of the two numbers is 1. Then, total number of cases
$$
=1 \times{ }^7 C_1=7
$$
Case II When smaller of two numbers is 2 . Then, total number of cases
$$
=1 \times{ }^6 C_1=6
$$
Case III When smaller of two numbers is 3 . Then, total number of cases
$$
=1 \times{ }^5 C_1=5
$$
Total favourable cases $=7+6+5=18$
Total case $={ }^8 \mathrm{C}_2=28$
$\therefore \quad$ Required probability $=\frac{18}{28}=\frac{9}{14}$
$$
=1 \times{ }^7 C_1=7
$$
Case II When smaller of two numbers is 2 . Then, total number of cases
$$
=1 \times{ }^6 C_1=6
$$
Case III When smaller of two numbers is 3 . Then, total number of cases
$$
=1 \times{ }^5 C_1=5
$$
Total favourable cases $=7+6+5=18$
Total case $={ }^8 \mathrm{C}_2=28$
$\therefore \quad$ Required probability $=\frac{18}{28}=\frac{9}{14}$
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