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 What is the counting interpretation of ({}^{n}C_r=\frac{n}{r}{}^{n-1}C_{r-1})?

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Answer and explanation

Derivations of formulas and their connections

Practice questions

01 What is the counting interpretation of ({}^{n}C_r=\frac{n}{r}{}^{n-1}C_{r-1})?

02 If (^{n}C_{r-1}:{}^{n}C_r=3:5), which expression is useful in the ratio derivation?

03 Which relation is most useful for identifying the largest term of ({}^{n}C_r)?

04 What is the formula for distributing (n) identical balls into (r) distinct boxes when empty boxes are allowed?

05 How does the formula change if (n) identical balls are distributed into (r) distinct boxes with every box non-empty?

06 The equality of ({}^{n+r-1}C_r) and ({}^{n+r-1}C_{n-1}) comes from which identity?

07 The count for seating (4) selected students in a row of (10) seats while leaving other seats empty is connected with which expression?

08 Why is the count for choosing (3) people from (7) into an ordered queue equal to (^{7}C_3\cdot3!)?

09 What is the formula for dividing (8) distinct objects into labelled groups of (3), (3), and (2)?

10 When (8) distinct objects are divided into unlabelled groups of sizes (3), (3), and (2), by what extra factor do we divide?

11 Which formula distributes (n) distinct objects into (r) distinct boxes when each object can go into one box?

12 Which inclusion-exclusion form counts onto distributions of (n) distinct objects into (r) distinct boxes?

13 (n) people are to be placed into (r) distinct rooms and rooms may be empty. Why is the count (r^n)?

14 Which condition gives the formula ({}^{n+r-1}C_{n}) for distributing (n) identical coins among (r) children?

15 What is the count of non-negative integer solutions of (x_1+x_2+x_3=12)?

16 If all (x_i\geq1) in (x_1+x_2+x_3+x_4=15), what is the count?

17 What is the formula for choosing (r) objects from (n) objects with exactly (s) special objects?

18 When choosing (r) objects from (n), if at least (1) special object is required and there are (a) special objects, what is the shortest expression?

19 From (10) books, (4) books are to be selected, and (2) fixed books must either both appear or both not appear. What is the count?

20 Choose (5) people from (9) people, but two particular people cannot be selected together. What is the count?

21 Arrange (8) people in a row so that (A) and (B) are not together. Which formula gives the count?

22 Arrange (6) boys and (4) girls in a row so that no two girls are together. What is the main step in the derivation?

23 A word is formed from (5) vowels and (7) consonants with no two vowels together. What is the vowel placement factor?

24 Under which condition does the repeated-object arrangement formula (\frac{n!}{p!q!}) apply?

25 Why is the denominator for distinct arrangements of (BANANA) equal to (3!2!)?

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