A teacher prepares a test. She gave $5$ questions of which $4$ need to be answered. Find the total number of ways of answering the questions if the first two questions have $3$ choices and the next three have $4$ choices.
2026-02-25 18:48:11.1772045291
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GRE Permutation and combination question
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Michael Biro has given us a great start - here's my work.
First, let's set this up in an easy to understand way.
Case 1: Question 1 is ignored, the rest are answered.
Case 2: Question 2 is ignored, the rest are answered.
Case 3: Question 3 is ignored, the rest are answered.
Case 4: Question 4 is ignored, the rest are answered.
Case 5: Question 5 is ignored, the rest are answered.
Case 6: All questions are answered.
As Michael stated, we are going to find the number of ways of answering the questions for each case and then add them up.
no. of ways in Case 1: 3*4*4*4 = 192
no. of ways in Case 2: 3*4*4*4 = 192
no. of ways in Case 3: 3*3*4*4 = 144
no. of ways in Case 4: 3*3*4*4 = 144
no. of ways in Case 5: 3*3*4*4 = 144
no. of ways in Case 6: 3*3*4*4*4 = 576
Which gives us a final answer of 1,392.
Split into cases based on whether a 3 choice question is skipped or a 4 choice question is skipped, and then add the answers.
If a 3 choice question is skipped, you have 2 choices for which one to skip, 3 choices to answer the remaining 3 choice question, and 4 choices for each of the 4 choice questions.
A similar analysis handles the other case.