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Business Mathematics and Statistics · Class 11 Commerce

Ch 2Algebra (Partial Fractions, Permutations, Combinations, Mathematical Induction, Binomial Theorem) — Class 11 Business Mathematics and Statistics, concept-first.

In business calculations we often work with a rational fraction — one polynomial divided by another, such as . Before such an expression can be split into simpler pieces, it must first be classified.

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Key concepts

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Partial Fraction Decomposition — Non-Repeated Linear Factors

When a proper rational fraction's denominator factorises into distinct (non-repeated) linear factors, such as , the fraction can always be written as a sum of simple fractions, one per factor, each with an unknown consta…

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Chapter contents

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Partial Fractions — Proper and Improper Rational Fractions

In business calculations we often work with a rational fraction — one polynomial divided by another, such as . Before such an expression can be split into simpler pieces, it must first be classified.

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Partial Fractions — Resolving into Partial Fractions

Once a rational fraction is proper, it can be broken into a sum of simpler fractions whose denominators are the individual factors of the original denominator.

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Permutations — The Fundamental Principle of Counting and Factorial Notation

Before permutations and combinations can be defined precisely, we need two basic tools.

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Permutations — nPr, Arrangements with Repetition, and Circular Permutations

A permutation is an arrangement of objects in a definite order — changing the order gives a different permutation.

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Combinations — nCr and Its Properties

A combination is a selection of objects where order does not matter — choosing A then B is the same selection as choosing B then A.

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Combinations — Choosing Between Permutation and Combination, and Business Applications

The single most useful question when solving a counting problem is: does the order of selection matter?

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Principle of Mathematical Induction — The Statement

Many results in business mathematics and statistics — such as summation formulas or divisibility properties — are claimed to hold for every natural number .

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Mathematical Induction — Applications to Summation and Divisibility

The Principle of Mathematical Induction is applied in this chapter to two standard classes of results.

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Binomial Theorem — Statement and the General Term

The Binomial Theorem gives a direct formula for expanding for a positive integer , without multiplying by itself times.

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Binomial Theorem — Middle Term(s) and Applications

Locating the middle term. The expansion of has terms.

Exercises

+Show 10 questions10 questions
  1. Q12Resolve $\dfrac{x^3}{(x-1)(x-2)}$ into partial fractions (reduce first, since the fraction is improper).Free
  2. Q13Resolve $\dfrac{x+7}{(x+1)(x-3)}$ into partial fractions.Free
  3. Q14Find the number of distinct arrangements of the letters of the word 'STATISTICS'.Free
  4. Q15In how many ways can 6 distinct people be seated around a circular table, if (i) clockwise and anticlockwise seatings are considered differe…Preview
  5. Q16Prove that $^{n}C_{r} + \, ^{n}C_{r-1} = \, ^{n+1}C_{r}$, and verify it numerically for $n=6,\,r=3$.Preview
  6. Q17A committee of 5 members is to be formed from 6 men and 4 women. In how many ways can this be done if the committee must contain exactly 3 m…Preview
  7. Q18Using the principle of mathematical induction, prove that $1^2+2^2+3^2+\cdots+n^2 = \dfrac{n(n+1)(2n+1)}{6}$ for all natural numbers $n$.Preview
  8. Q19Using the principle of mathematical induction, prove that $7^{n} - 3^{n}$ is divisible by $4$ for every natural number $n$.Preview
  9. Q20Use the binomial theorem to find the approximate value of $(1.02)^{5}$, correct to three decimal places, keeping terms only up to the $y^2$…Preview
  10. Q21Find the term independent of $x$ in the expansion of $\left(x^2 - \dfrac{2}{x}\right)^{9}$.Preview

Sample & Board Papers

Sample papers and previous-year board questions for this subject.

+Show 27 questions27 questions
  1. Q1The total number of 9 digit numbers which have all different digits is : (a) $10 \times 10!$ (b) $10!$ (c) $9!$ (d) $9 \times 9!$Preview
  2. Q2Resolve into partial fraction : $\dfrac{4}{x^2-1}$.Preview
  3. Q3If $(n+2)C_n = 45$, find $n$.Preview
  4. Q4(a) Show that the middle term in the expansion of $(1+x)^{2n}$ is $\dfrac{1 \cdot 3 \cdot 5 \ldots (2n-1)\, 2^n x^n}{n!}$. OR (b) Find out t…Preview
  5. Q5The value of $n$, when $^{n}P_2 = 20$ is : (a) $5$ (b) $3$ (c) $4$ (d) $6$Preview
  6. Q6The number of permutation of $n$ different things taken $r$ at a time, when the repetition is allowed is : (a) $\dfrac{n!}{(n-r)!}$ (b) $r^{…Preview
  7. Q7Find how many four letter words can be formed from the letters of the word "LOGARITHMS" (letters not repeated and words are with or without…Preview
  8. Q8Out of 7 consonants and 4 vowels, how many words of 3 consonants and 2 vowels can be formed ?Preview
  9. Q9(a) A cricket team of 11 players is to be formed from 16 players including 4 bowlers and 2 wicket-keepers. In how many different ways can a…Preview
  10. Q10The number of ways in selecting 4 players out of 5 is : (a) 25 (b) $4!$ (c) 5 (d) 20Preview
  11. Q11Sum of the binomial coefficients is : (a) $2n$ (b) $2^n$ (c) $n+17$ (d) $n^2$Preview
  12. Q12In how many ways 7 pictures can be hung from 5 picture nails on a wall ?Preview
  13. Q13Find the rank of the word 'CHAT' in dictionary.Preview
  14. Q14(a) Find the term independent of $x$ in the expansion of $\left(2x^2 + \dfrac{1}{x}\right)^{12}$. OR (b) If the demand for a commodity $x$ i…Preview
  15. Q15(a) By mathematical Induction, prove that $1^3 + 2^3 + 3^3 + \ldots + n^3 = \dfrac{n^2(n+1)^2}{4}$ for all $n \in N$. OR (b) Verify the cont…Preview
  16. Q16The number of permutation of $n$ different things taken $r$ at a time, when the repetition is allowed is : (a) $\dfrac{n!}{(n-r)!}$ (b) $r^n…Preview
  17. Q17If clockwise and anticlockwise circular permutations are considered to be same, the number of circular permutation of $n$ objects taken all…Preview
  18. Q18Find the value of $^{100}C_{99}$ : (a) $1$ (b) $100$ (c) $0$ (d) $99$Preview
  19. Q19If $^nC_4={}^nC_6$, find $^{12}C_n$.Preview
  20. Q20Find the $5^{\text{th}}$ term in the expansion of $(x-2y)^{13}$.Preview
  21. Q21Find the rank of the word "TABLE" in English dictionary.Preview
  22. Q22(a) By Mathematical Induction, prove that $1^2+2^2+3^2+\dots+n^2=\dfrac{n(n+1)(2n+1)}{6}$, for all $n\in N$. OR (b) If $\sin y=x\sin(a+y)$,…Preview
  23. Q23Thirteen guests have participated in a dinner. The number of handshakes that happened in the dinner is : (a) $286$ (b) $715$ (c) $13$ (d) $7…Preview
  24. Q24The number of ways 8 identical flowers can be arranged in a ring is ________. (a) $\dfrac{7}{2}$ (b) $8!$ (c) $\dfrac{7!}{2}$ (d) $\dfrac{8!…Preview
  25. Q25Find the rank of the word 'RANK' in dictionary.Preview
  26. Q26How many five digit telephone numbers can be constructed using the digits 0 to 9 if each number starts with 67 with no digit appearing more…Preview
  27. Q27Resolve into partial fractions. $\dfrac{2x-1}{x^2-5x+6}$Preview

More questions

+Show 11 questions11 questions
  1. Example 1Resolve $\dfrac{3x+5}{(x-1)(x+2)}$ into partial fractions.Free
  2. Example 2Resolve $\dfrac{x+3}{(x-1)^2(x+2)}$ into partial fractions.Free
  3. Example 3Resolve $\dfrac{2x+1}{(x-1)(x^2+1)}$ into partial fractions.Free
  4. Example 4Using the fundamental principle of counting, find how many 4-digit numbers (with no digit repeated) can be formed using the digits $1,2,3,4,…Preview
  5. Example 5Simplify $\dfrac{(n+2)!}{n!} - \dfrac{n!}{(n-1)!}$, and hence evaluate it for $n=5$.Preview
  6. Example 6(i) Evaluate $^{7}P_{3}$. (ii) In how many ways can 3 books be selected and arranged, in order, on a shelf from a collection of 7 different…Preview
  7. Example 7(i) Evaluate $^{10}C_{4}$. (ii) A business committee of 4 members is to be selected from 10 candidates. In how many ways can this be done?Preview
  8. Example 8Using the principle of mathematical induction, prove that $1+2+3+\cdots+n = \dfrac{n(n+1)}{2}$ for all natural numbers $n$.Preview
  9. Example 9Expand $(x+2)^5$ using the binomial theorem.Preview
  10. Example 10Find the 5th term in the expansion of $(2x-y)^{8}$.Preview
  11. Example 11Find the middle term in the expansion of $\left(x + \dfrac{1}{x}\right)^{8}$.Preview