Multiple choice

From a box containing 10 cards numbered 1 to10, four cards are drawn together. The probability that their sum is odd is

  1. $\displaystyle \frac{1}{2}$
  2. $\displaystyle \frac{11}{21}$
  3. $\displaystyle \frac{10}{21}$
  4. $\displaystyle \frac{1}{21}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Total ways to choose 4 cards from 10 is 10C4 = 210. For the sum to be odd, we need an odd number of odd cards (1 or 3). Odd cards: {1,3,5,7,9} (5), Even cards: {2,4,6,8,10} (5). Case 1 (1 odd, 3 even): 5C1 * 5C3 = 5 * 10 = 50. Case 2 (3 odd, 1 even): 5C3 * 5C1 = 10 * 5 = 50. Total favorable = 100. Probability = 100/210 = 10/21.

AI explanation

The four cards sum to an odd number only if an odd amount of the drawn cards feature odd digits, meaning either 1 or 3 cards are odd. The total ways to choose 4 cards from 10 is 10C4, which equals 210. The favourable ways are (5C1 * 5C3) + (5C3 * 5C1), which equals 100. The probability of drawing a sum that is odd is therefore 100/210, which simplifies to 10/21.

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