Mathematics

Derivations of formulas and their connections

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.

Practice questions

01 In a line of (n) people, if exactly (k) people are between (A) and (B), what is the count?

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Answer and explanation

Derivations of formulas and their connections

Practice questions

01 In a line of (n) people, if exactly (k) people are between (A) and (B), what is the count?

02 In a line of (11) people, if exactly (4) people are between (A) and (B), what is the count?

03 (10) people are seated around a round table and exactly (3) people lie clockwise between (A) and (B). What is the count?

04 (9) people are seated around a round table and (A) and (B) are not adjacent. What is the count?

05 What is the count for seating (8) men and (8) women alternately around a round table?

06 For necklace arrangements of (10) distinct beads where rotations are the same but reflections are different, what is the count?

07 What is the count for bracelet arrangements of (10) distinct beads?

08 Using digits (0,1,2,3,4,5,6,7,8,9) without repetition, (6)-digit even numbers are formed. What is the count when (0) is the last digit?

09 Using digits (0) to (9) without repetition, what is the count for (6)-digit even numbers with a non-zero even last digit?

10 Using digits (1,2,3,4,5,6,7,8) with repetition allowed, if exactly (3) even digits occur in (6)-digit numbers, what is the count?

11 Length (8) strings are formed from (5) symbols and every symbol appears at least once. Which inclusion-exclusion form gives the count?

12 Length (9) strings are formed from (6) symbols and exactly (4) distinct symbols are used. Which count is correct?

13 In length (r) strings with exactly (s) distinct symbols used, what does (s!,S(r,s)) count?

14 What is the coefficient of (a^2b^3c^1d^4) in ((a+b+c+d)^{10})?

15 Which expression gives the coefficient of (x^3) in ((1+x+x^2)^{10})?

16 Which option correctly lists cases for the coefficient of (x^4) in ((1+x+x^2)^n)?

17 Which method is used to separate coefficients with indices (0,3,6,\ldots) in ((1+x)^n)?

18 If ({}^{n}C_{r+1}>{}^{n}C_r), which condition is correct?

19 If ({}^{n}C_{r+1}<{}^{n}C_r), which inequality is correct?

20 If ({}^{n}C_{r}= {}^{n}C_{r+6}) and the indices are unequal, which relation is correct?

21 If ({}^{30}C_{3r-2}={}^{30}C_{r+8}) and the lower indices are unequal, what is (r)?

22 If ({}^{n}P_5=15{}^{n}P_4), what is the value of (n)?

23 If (\frac{{}^{n}C_{r+1}}{{}^{n}C_r}=\frac{4}{5}), which relation is correct?

24 If (\frac{{}^{n}C_{r}}{{}^{n}C_{r-1}}=\frac{7}{3}), which relation is formed?

25 A team of (5) is formed from (12) candidates and at least (2) of (A,B,C) are selected. Which count is correct?

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