Selections with at least or at most conditions (split into cases)

This type of question, asked 6 times: May 2022 S1, 3 May 2023 S2, 15 May 2023 S2, 9 May 2024 S2, 10 May 2024 S1, 14 May 2024 S1. All 6 questions below, oldest first.

Try each one first, then open its answer.

Question 1
May 2022 S1
128 $If a question paper consists of 11 questions divided into two sections I and II. Section I consists of 6 questions and section II consists of 5 questions, then the number of different ways a student can select 6 questions, taking at least 2 questions from each section, is$
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Question 2
3 May 2023 S2
130 $m$
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Question 3
15 May 2023 S2
109 $There are 3 sections in a question paper and each section contains 5 questions. A candidate has to answer a total of 5 questions, choosing at least one question from each section. Then the number of ways, in which the candidate can choose the questions, is$
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Question 4
9 May 2024 S2
110 $A linguistic club consists of 6 girls and 4 boys. A team of 4 members to be selected including selection of a Captain (from among these 4 members). If the team has to include at most one boy, the number of ways of selecting the team is$
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Question 5
10 May 2024 S1
132 $A group consists of 8 boys and 5 girls, then the number of committees of 5 persons that can be formed, if the committee consists of at least 2 girls and at most 2 boys, is$
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Question 6
14 May 2024 S1
118 $A linguistic club consists of 6 girls and 4 boys. A team of 4 members is to be selected from this group including the selection of a leader (from among these 4 members) for the team. If the team has to include at most one boy, the number of ways of selecting the team is$
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Selections with at least or at most conditions (split into cases)

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6