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 (8) people are seated around a round table and (A) and (B) are not adjacent. Which count is correct?

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Answer and explanation

Derivations of formulas and their connections

Practice questions

01 (8) people are seated around a round table and (A) and (B) are not adjacent. Which count is correct?

02 (7) couples are seated around a round table and every couple stays together. What is the count?

03 What is the count for seating (6) men and (6) women alternately around a round table?

04 What is the count for arranging (9) distinct beads in a bracelet?

05 Using (0,1,2,3,4,5,6,7,8) without repetition, what is the count for (5)-digit even numbers when (0) is the last digit?

06 Using digits (0,1,2,3,4,5,6,7,8) without repetition, what is the count for (5)-digit even numbers with a non-zero even last digit?

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

08 Length (6) strings are formed from (4) symbols and every symbol appears at least once. Which count is correct?

09 Length (7) strings are formed from (5) symbols and exactly (3) distinct symbols are used. Which form is correct?

10 For length (r) strings from (n) distinct symbols with exactly one repeated symbol and all others distinct, what is the core expression?

11 What is the coefficient of (a^p b^q c^r d^s) in ((a+b+c+d)^n), if (p+q+r+s=n)?

12 What is the coefficient of (x^4y^3z^2) in ((x+y+z)^9)?

13 How will the coefficient of (x^2) in ((1+x+x^2)^n) be derived?

14 What is the coefficient of (x^2) in ((1+x+x^2)^8)?

15 What advanced technique separates the sum of coefficients in ((1+x)^n) whose indices are divisible by (3)?

16 Which ratio is used in the ratio test for the maximum term of ({}^{n}C_r)?

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

18 If ({}^{n}C_{r}= {}^{n}C_{r+4}) and the indices are not equal, what is the relation between (r) and (n)?

19 If ({}^{24}C_{2r-1}={}^{24}C_{r+8}) and the lower indices are unequal, what is (r)?

20 If ({}^{n}P_4=12{}^{n}P_3), what is the value of (n)?

21 If (\frac{{}^{n}C_{r+1}}{{}^{n}C_r}=\frac{3}{4}), which relation is correct?

22 If (\frac{{}^{n}C_{r}}{{}^{n}C_{r-1}}=\frac{5}{2}), which relation is formed?

23 When choosing (r) objects from (n) distinct objects, at least (2) special objects are required and there are (s) special objects. Which complement expression is correct?

24 A committee of (5) is formed from (10) students, and (A) and (B) are either both included or both excluded. What is the count?

25 (4) players are chosen from (9) players, and (A) and (B) are not selected together. Which count is correct?

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