<p>Two numbers are chosen from {1, 2, 3, 4, 5, 6, 7, 8} one after another without replacement. Then the probability that</p><p>(1) the smaller value of two is less than 3 is 13/28<br>(2) the bigger value of two is more than 5 is 9/14<br>(3) product of two number is even is 11/14<br>(4) none of these</p>
<p>the smaller value of two is less than 3 is 13/28</p>
<p>the bigger value of two is more than 5 is 9/14</p>
<p>product of two number is even is 11/14</p>
<p>none of these</p>
Step-by-Step Solution
Key Concept: When selecting two numbers without replacement, count favorable outcomes carefully by considering ordered pairs (first, second) where order matters for 'without replacement', then reduce by total possible ordered selections. For conditions like 'smaller < 3' or 'larger > 5', identify which pairs satisfy the condition systematically.
<p><strong>Step 1: Total outcomes</strong><br>Two numbers chosen one after another without replacement = 8 × 7 = 56 ordered pairs</p><p><strong>Step 2: Check condition (1) - Smaller value < 3</strong><br>Smaller value must be 1 or 2.<br>If first number is 1: pair with any of {2,3,4,5,6,7,8} = 7 pairs<br>If first number is 2: pair with any of {1,3,4,5,6,7,8} = 7 pairs<br>If second number is 1: first from {2,3,4,5,6,7,8} but smaller must be 1 = covered above<br>If second number is 2: first from {1,3,4,5,6,7,8} but smaller must be 2 = only if first > 2: {3,4,5,6,7,8} = 6 pairs<br>Total = 7 + 7 + 6 = 20 pairs<br>Probability = 20/56 = 5/14 ≠ 13/28. <strong>Statement (1) is FALSE</strong></p><p><strong>Step 3: Check condition (2) - Larger value > 5</strong><br>Larger value is in {6,7,8}.<br>Complement easier: larger ≤ 5 means both numbers from {1,2,3,4,5}<br>Favorable pairs = 5 × 4 = 20<br>Unfavorable pairs = 56 - 20 = 36<br>Probability = 36/56 = 9/14. <strong>Statement (2) is TRUE</strong></p><p><strong>Step 4: Check condition (3) - Product is even</strong><br>Product is even if at least one number is even.<br>Complement: product is odd only if both numbers are odd.<br>Odd numbers: {1,3,5,7} = 4 numbers<br>Both odd = 4 × 3 = 12 pairs<br>At least one even = 56 - 12 = 44 pairs<br>Probability = 44/56 = 11/14. <strong>Statement (3) is TRUE</strong></p><p><strong>Step 5: Evaluate options</strong><br>Statement (1) is FALSE<br>Statements (2) and (3) are both TRUE<br>Since multiple statements are true, the answer is <strong>(4) none of these</strong></p>
Correct Answer: C