<p>In an exciting finish to an one-day cricket match between India and Pakistan, the Indians require 10 runs in the last 3 balls to win. If any one of the scores 0, 1, 2, 3, 4, 6 can be made from a ball and no wides or no-balls are bowled then in how many different sequences can the batsmen make exactly 10 runs?</p>
Step-by-Step Solution
Key Concept: Find all ordered partitions of 10 into exactly 3 parts, where each part must be from the set {0,1,2,3,4,6}. This requires systematic enumeration of valid combinations and then counting their permutations.
<p><strong>Step 1:</strong> We need three scores from {0,1,2,3,4,6} that sum to exactly 10. List all unordered combinations:</p><p>• (0,4,6): sum = 10 ✓</p><p>• (1,3,6): sum = 10 ✓</p><p>• (2,2,6): sum = 10 ✓</p><p>• (2,3,5): Not valid (5 not allowed)</p><p>• (3,3,4): sum = 10 ✓</p><p>• (4,4,2): Already counted as (2,4,4) ✓</p><p>• (1,4,5): Not valid (5 not allowed)</p><p><strong>Step 2:</strong> Count permutations (sequences) for each combination:</p><p>• (0,4,6): All different → 3! = 6 sequences</p><p>• (1,3,6): All different → 3! = 6 sequences</p><p>• (2,2,6): Two 2's and one 6 → 3!/2! = 3 sequences</p><p>• (3,3,4): Two 3's and one 4 → 3!/2! = 3 sequences</p><p>• (2,4,4): Two 4's and one 2 → 3!/2! = 3 sequences</p><p><strong>Step 3:</strong> Total sequences = 6 + 6 + 3 + 3 + 3 = 21</p><p>∴ Answer: <strong>21</strong></p>
Correct Answer: 21