In a class, every student shook hands once with every other student. If the total number of handshakes is 190, how many students are there?
Answer and explanation
Correct answer: 20
If there are n students, each handshake corresponds to choosing an unordered pair of distinct students. Therefore the number of handshakes is C(n, 2) = n(n − 1)/2. Setting this equal to 190 gives n(n − 1) = 380, or n^2 − n − 380 = 0. Factoring yields (n − 20)(n + 19) = 0, so n = 20 or n = −19. A number of students cannot be negative, hence n = 20. Option C is correct. The negative root is rejected by the context, while 18, 19, and 21 do not satisfy the pair-count formula: for 20 students, 20 × 19 / 2 = 190 exactly. This is a standard quadratic-equation application based on counting pairs.
Frequently asked questions
What is the correct answer to this question?
20
Why is this the correct answer?
If there are n students, each handshake corresponds to choosing an unordered pair of distinct students. Therefore the number of handshakes is C(n, 2) = n(n − 1)/2. Setting this equal to 190 gives n(n − 1) = 380, or n^2 − n − 380 = 0. Factoring yields (n − 20)(n + 19) = 0, so n = 20 or n = −19. A number of students cannot be negative, hence n = 20. Option C is correct. The negative root is rejected by the context, while 18, 19, and 21 do not satisfy the pair-count formula: for 20 students, 20 × 19 / 2 = 190 exactly. This is a standard quadratic-equation application based on counting pairs.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Quadratic Equations. Topic: Word Problems and Applications.
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