Permutations and Combinations
Class 11 Mathematics · Chapter 6
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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: