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SubjectsMathematics
सूत्रों की व्युत्पत्तियाँ और उनके पारस्परिक संबंध
In Class 11 Mathematics, this topic explains how the formulas in Permutations and Combinations are derived and how they are connected. Students learn the meaning of factorial notation, develop the formulas for nPr and nCr from counting principles, and understand why nPr = r! nCr and nCr = nC(n−r). The topic also shows when to use arrangements or selections, helping students apply these relationships logically instead of relying on memorized formulas.
Correct answer: A. (3!)
Explanation: The three red balls are identical so their (3!) internal orders are not different. In exams divide by the factorial of the repeated group only.
Correct answer: B. (\frac{8!}{2!3!})
Explanation: To remove internal orders of two repeated groups divide by (2!3!). In exams put the factorial of each repeated letter in the denominator.
Correct answer: A. Because one person is fixed to remove rotations
Explanation: A mere rotation in a circle does not make a new seating. In exams use the one fixed method in circular arrangement.
Correct answer: C. (\frac{(n-1)!}{2})
Explanation: First removing rotations gives ((n-1)!) and then same reflection makes us divide by (2). In exams check both rotation and reflection in necklaces.
Correct answer: A. When (r=0) or (r=1)
Explanation: Because (r!) is (1) only for (r=0) and (r=1). In exams check (^{n}P_r=^{n}C_r r!) for equality.
Correct answer: A. Because in combination (r!) orders are treated as one selection
Explanation: Permutation counts many orders of the same group. In exams keep the difference between (P) and (C) clear when (r>1).
Correct answer: A. When extra repeated orders must be removed
Explanation: When making combination from permutation the same selection is counted (r!) times. In exams use division principle when overcounting appears.
Correct answer: B. Multiplication principle
Explanation: Each shirt can pair with (4) pants so there are (3\times4) choices. In exams multiply when independent choices are needed together.
Correct answer: B. Addition principle
Explanation: This is an or situation so there are (5+6) choices. In exams look for the addition principle when the word or appears.
Correct answer: D. Rank
Explanation: Changing rank changes the result so order is important. In exams use permutation when rank or post appears.
Correct answer: B. Number of unchosen objects
Explanation: After choosing (r) objects (n-r) objects remain. In exams remember this meaning for complement counting.
Correct answer: C. 7
Explanation: In permutations, \(^{n}P_r=n(n-1)(n-2)\cdots(n-r+1)\), so the last factor is \(n-r+1\). Here, \(10-4+1=7\), so the correct answer is 7. The value 6 would result from incorrectly omitting \(+1\). Exam tip: write \(n-r+1\) first and then substitute the values.
Correct answer: A. Selecting a president and a secretary from four students
Explanation: President and secretary are distinct posts, so interchanging the same two students gives a different outcome; hence order matters and permutations apply. In a committee, order does not matter. Exam tip: look for roles or positions to identify permutations.
Correct answer: A. Choosing no object
Explanation: Choosing no object gives one empty selection. In exams treat the empty case as valid count (1).
Correct answer: B. Arranging all (n) objects in order
Explanation: When (r=n) all objects are being arranged so the count is (n!). In exams treat full arrangement as factorial.
Correct answer: A. chosen group and rejected group
Explanation: Each selection has a unique rejected group. In exams understand the identity by forming a complementary pair.
Correct answer: A. Because initially (n) objects can be counted by arranging in order
Explanation: (n!) creates a larger ordered count which is corrected by (r!(n-r)!). In exams understand the overcounting correction.
Correct answer: B. (^{5}C_2)
Explanation: A line segment is an unordered pair of two points. In exams do not count the order of endpoints in a segment.
Correct answer: B. (^{5}P_2)
Explanation: In a directed arrow changing start and end changes the arrow. In exams use ordered pair or permutation when direction exists.
Correct answer: A. 7 + 9 = 16, so they are complementary selections
Explanation: The governing idea is complementary selection, represented by the combination identity ⁿCᵣ = ⁿCₙ₋ᵣ. From a collection of 16 objects, every selection of 7 objects determines exactly one complementary group of 9 objects that was not selected. This one-to-one correspondence proves that the number of 7-element selections equals the number of 9-element selections. Algebraically, 16−7=9, or equivalently 7+9=16, so ¹⁶C₇=¹⁶C₉. Hence option A is correct. Option B is irrelevant because multiplication is not the condition in the symmetry identity. Option C is false because the lower indices are different, and option D confuses combinations, where order is ignored, with permutations, where order matters.
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