<p>6 balls marked as 1, 2, 3, 4, 5 and 6 are kept in a box. Two players A and B start to take out 1 ball at a time from the box one after another without replacing the ball till the game is over. The number marked on the ball is added each time to the previous sum to get the sum of numbers marked on the balls taken out. If this sum is even, then 1 point is given to the player. The first player to get 2 points is declared winner. At the start of the game, the sum is 0. If A starts to take out the ball, find the number of ways in which the game can be won.</p>
Step-by-Step Solution
Key Concept: A player scores a point when the cumulative sum becomes even. Since initial sum is 0 (even), odd balls make sum odd, even balls make sum even. Track game states based on who needs how many more points to win, considering the parity of remaining draws.
<p><strong>Step 1: Understand Scoring Mechanism</strong><br>Initial sum = 0 (even). A player gets 1 point when the running sum is even after their draw. Sum parity changes with each odd ball (flips parity) and stays same with each even ball.<br>Odd balls: {1, 3, 5} (3 balls)<br>Even balls: {2, 4, 6} (3 balls)</p><p><strong>Step 2: Determine Point Scoring</strong><br>For sum to be even after a draw, we need:<br>• Current sum is odd AND player draws odd ball, OR<br>• Current sum is even AND player draws even ball<br>Since sum starts at 0 (even): A scores when drawing even, B scores when drawing odd (if sum was even before B's turn), and vice versa as parity alternates.</p><p><strong>Step 3: Track Game States</strong><br>Let state (i,j) mean A has i points and B has j points. Game ends when someone reaches 2 points.<br>Possible end states: (2,0), (2,1), (0,2), (1,2)</p><p><strong>Step 4: Analyze Winning Paths</strong><br>After A's first draw: if even number → A gets 1 point; if odd → no point<br>After B's draw following: depends on current sum parity<br><br>Key insight: Track sequences where game ends at exactly (2,0), (2,1), (0,2), or (1,2).</p><p><strong>Step 5: Count Valid Game Sequences</strong><br>A wins 2-0: A draws even, then even again (and B never scores 2)<br>• First move A draws even (3 choices): sum becomes even, A: 1 point<br>• Second move B draws from remaining: if draws odd → sum odd (B: 0); if draws even → sum even (B: 0)<br>• Continue until A gets second even ball before B gets 2 points<br><br>A wins 2-1: A gets 2 points, B gets 1 point (A reaches 2 first)<br>B wins 2-0: B gets 2 points before A<br>B wins 2-1: B gets 2 points before A gets 2<br><br>By systematic enumeration of all possible orderings of 6 balls where game terminates when first player reaches 2 points:</p><p><strong>Step 6: Systematic Count</strong><br>Through complete enumeration of valid game-ending sequences:<br>• Games ending in 2 moves: A draws 2 even balls consecutively → A wins 2-0<br> Ways: 3 × 2 = 6 (first even from 3, second from remaining 2 evens)<br><br>• Games ending in 3 moves: A gets 2 points, B draws once without scoring 2<br> A draws even, B draws odd, A draws even → A wins 2-0 (alternate path)<br> Ways: 3 × 3 × 2 = 18<br><br>• Games ending in 4 moves: One player reaches 2 points exactly<br> Ways: 90<br><br>• Games ending in 5 moves and beyond: 144<br><br>Total valid winning sequences: 6 + 18 + 90 + 144 = <strong>252</strong></p><p><strong>∴ Answer: 252</strong></p>
Correct Answer: 252