## Solve Lesson 49 Page 18 Math Workbook 10 – Kite>

Topic

A test for good students in grade 10 in Mathematics consists of 5 questions selected from 15 comprehension sentences, 10 low application sentences and 5 high application sentences. A test is said to be good if there are all three levels in the test, and the number of comprehension questions is not less than 2. How many good test questions can be made?

Solution method – See details

Divide the action into independent cases (in which the number of sentences is understood from 2 or more and there are 3 levels) and then calculate the number of choices by applying the rules of multiplication and combination.

 Cases Understanding Low usage High usage first 2 first 2 2 2 2 first 3 3 first first

Detailed explanation

Because the test has no less than 2 comprehension questions and 3 levels, the following cases occur:

Case 1: The test has 2 comprehension sentences, 1 low application sentence and 2 high application sentences.

Then we have \(C_{15}^2.C_{10}^1.C_5^2 = 10500\) how to choose the topic

Case 2: The test has 2 comprehension sentences, 2 low application sentences and 1 high application sentence.

Then we have \(C_{15}^2.C_{10}^2.C_5^1 = 23625\) how to choose the topic

Case 3: The test has 3 comprehension sentences, 1 low application sentence and 1 high application sentence.

Then we have \(C_{15}^3.C_{10}^1.C_5^1 = 22750\) how to choose the topic

So the number of good exam questions that can be selected is: 22 750 + 23 625 + 10 500 = 56 875 questions