What went wrong when the king rode under the arch?

It was too low and knocked off his crown

How does the chain of blame unfold in the poem?

Each accused person blames another, ending with the king's own changes

State the fundamental principle of counting.

If an event can occur in \(m\) ways and a following event in \(n\) ways, the total number of ways the events occur in order is \(m\times n\); this extends to any finite number of events.

Write the formula for \({}^{n}P_r\) and the meaning of \(0!\).

\({}^{n}P_r=\dfrac{n!}{(n-r)!}\) for \(0\le r\le n\), and by convention \(0!=1\) so that \({}^{n}P_0=1\) and \({}^{n}P_n=n!\).

How many 3-letter words (no repetition) come from ROSE's letters?

Choosing ordered letters from 4: \({}^{4}P_3=4\times3\times2=24\) words.

Distinguish a permutation from a combination.

A permutation counts arrangements where order matters; a combination counts selections where order does not matter. \({}^{n}P_r={}^{n}C_r\cdot r!\).

State the relation \({}^{n}C_r\) and \({}^{n}C_{n-r}\).

\({}^{n}C_r={}^{n}C_{n-r}\), since choosing \(r\) objects is equivalent to rejecting \(n-r\).

Prove \({}^{n}P_r={}^{n}C_r\cdot r!\).

Each combination of \(r\) objects can be ordered in \(r!\) ways, producing all permutations, so \({}^{n}P_r={}^{n}C_r\cdot r!\).

How many arrangements has the word ALLAHABAD (9 letters, 4 A, 2 L)?

\(\dfrac{9!}{4!\,2!}=7560\) arrangements.

In how many ways can a committee of 3 be chosen from 5 people?

\({}^{5}C_3=\dfrac{5!}{3!\,2!}=10\) ways.

How many 4-letter words come from EQUATION using each letter once?

All 8 letters are distinct, so 4-letter words \(={}^{8}P_4=8\times7\times6\times5=1680\).

Permutations and Combinations

Class 11 Mathematics · Chapter 6

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Every formula of Ch 1–14, with its condition

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Every formula of Ch 1–14, with its condition

Marks booster + cheatsheet: key concepts in 2 lines each, common traps
25 marks: Unit II, Ch 4–8
23 marks: Unit I, Ch 1–3
Common traps, corrected
Conic table and standard values
Graphs plotted from the book's own tables

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Summary

This chapter develops counting techniques. The fundamental principle of counting (multiplication principle) states that if one event can occur in \(m\) ways and a following event in \(n\) ways, the two together occur in \(m\times n\) ways. A permutation is an arrangement in a definite order; the number of permutations of \(n\) distinct objects taken \(r\) at a time is \({}^{n}P_r=\dfrac{n!}{(n-r)!}\), where \(n!=1\times2\times\cdots\times n\) and \(0!=1\). When repetition is allowed there are \(n^r\) arrangements, and when some objects repeat (\(p_1,p_2,\dots\) of each kind) the count is \(\dfrac{n!}{p_1!\,p_2!\cdots}\). A combination is a selection where order does not matter; the number of combinations is \({}^{n}C_r=\dfrac{n!}{r!\,(n-r)!}\). Key relations include \({}^{n}P_r={}^{n}C_r\cdot r!\), \({}^{n}C_r={}^{n}C_{n-r}\), \({}^{n}C_0={}^{n}C_n=1\) and the Pascal rule \({}^{n}C_r+{}^{n}C_{r-1}={}^{n+1}C_r\). Selecting \(r\) objects out of \(n\) is the same as rejecting the remaining \(n-r\), which explains the symmetry of \({}^{n}C_r\). These ideas count three-digit numbers, signals from flags, words formed from the letters of a given word, committees with conditions, card hands drawn from a deck, and arrangements where some objects must stay together or apart.

Fundamental principle of countingFactorial notationPermutations of distinct objectsPermutations with repetition and with like objectsCombinationsRelations between \({}^{n}P_r\) and \({}^{n}C_r\)

Key terms

Fundamental principle of counting

If one event occurs in \(m\) ways and another in \(n\) ways, both occur in \(m\times n\) ways.

Factorial

\(n!=1\times2\times3\times\cdots\times n\), with \(0!=1\).

Permutation

An ordered arrangement; \({}^{n}P_r=\dfrac{n!}{(n-r)!}\).

Combination

A selection without order; \({}^{n}C_r=\dfrac{n!}{r!\,(n-r)!}\).

Permutations with repetition

\(n^r\) arrangements when repetition of objects is allowed.

Like objects

With \(p_1,p_2,\dots\) repeated objects, arrangements \(=\dfrac{n!}{p_1!\,p_2!\cdots}\).

Extra questions & answers

Try one from this chapter

By the fundamental principle of counting, if one event occurs in \(m\) ways and another in \(n\) ways, both occur in:

Flashcards & interactive practice