CBSE Class 11-science Answered

 There are 10 questions in the question paper, each question can be answered in 2 ways (Either YES or NO).

(1)    Therefore, the number of ways that he answers all questions = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 2¹⁰ = 1024.

(2)    To attempt at least one question, he may attempt either of 1, 2, 3, 4, 5, 6, 7, 8, 9 and      10 questions.

Out of 10 questions, 1 question can be selected in ¹⁰C₁ ways.

Each of these questions can be answered in 2¹ ways.

So, 1 question can be answered in ¹⁰C₁ × 2¹ ways.

Out of 10 questions, 2 questions can be selected in ¹⁰C₂ ways.

Any of these 2 questions can be answered in 2² ways.

So, 2 questions can be answered in ¹⁰C₂ × 2² ways.

Similarly, the number of ways in which he answers at least one question

= ¹⁰C₁ × 2¹ + ¹⁰C₂ × 2² + ¹⁰C₃ × 2³ + ¹⁰C₄ × 2⁴ + ¹⁰C₅ × 2⁵ + ¹⁰C₆ × 2⁶ + ¹⁰C₇ × 2⁷ + ¹⁰C₈ × 2⁸ + ¹⁰C₉ × 2⁹ + ¹⁰C₁₀ × 2¹⁰

= (¹⁰C₀ × 2⁰ + ¹⁰C₁ × 2¹ + ¹⁰C₂ × 2² + ¹⁰C₃ × 2³ + ¹⁰C₄ × 2⁴ + ¹⁰C₅ × 2⁵ + ¹⁰C₆ × 2⁶ + ¹⁰C₇ × 2⁷ + ¹⁰C₈ × 2⁸ + ¹⁰C₉ × 2⁹ + ¹⁰C₁₀ × 2¹⁰) - ¹⁰C₀ × 2⁰

= (2 + 1)¹⁰ - 1

= 3¹⁰ - 1

(3)    No question can be answered in only 1 way.

Some questions i.e. at least one question can be answered in (3¹⁰ - 1) ways.

So, no question or some questions can be answered in (3¹⁰ - 1) + 1 = 3¹⁰ ways.