Multiple choice

The number of ways in which $5$ balls can be selected from a bag containing $5$ identical and $5$ different balls is

  1. $24$
  2. $30$
  3. $32$
  4. $36$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

We need to select 5 balls from 5 identical and 5 different balls. Let i be the number of identical balls chosen (0 to 5) and d be the number of different balls chosen (5-i). The number of ways is the sum of combinations of 5 different balls taken (5-i) at a time: 5C0 + 5C1 + 5C2 + 5C3 + 5C4 + 5C5 = 2^5 = 32.

AI explanation

Let k be the number of identical balls selected, where k ranges from 0 to 5, meaning the number of different balls selected is 5 minus k. The identical balls can only be chosen in exactly 1 way for any value of k. The 5 minus k different balls must be chosen from the 5 available different balls, which equals 5C(5 minus k). Summing the combinations for all possible values of k gives 5C5 plus 5C4 plus 5C3 plus 5C2 plus 5C1 plus 5C0, which is the sum of all binomial coefficients in the fifth row. This sum evaluates to 2 to the power of 5, resulting in 32.

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