Counting Rules
Multiplication
Suppose one starts his journey from place X and has to reach place Z via a different place Y. For Y, there are three means of transport  bus, train and aeroplane  from X. From Y, the aeroplane service is not available for Z. Only either by a bus or by a train can one reach Z from Y. Also, there is no direct bus or train service fro Z from X. We want to know the maximum possible no. of ways by which one can reach Z from X.
Sol:
= 3 × 2
= 6
If a work A can be done in m ways and another work B can be done in n ways and C is the final work which is done only when both A and B are done, then the no. of ways of doing the final work ?
Sol :
C = m × n
C = 3 × 2 = 6
From each group of two persons we have one handshake.
Case 1 : Total no. of handshakes among the group of 42 men
^{42}C_{2 }= 42!/2! (422)! = 21 × 41 = 861
Case 2 : Total no. of handshakes among the group of 16 women
^{16}C_{2 }= 16!/2! (162)! = 8 × 15 = 120
so maximum no. of handshakes = 861 + 120 = 981.
Problems and Solutions
Ques 1. How many numbers of five digits can be formed with the digits 1,3,5 7 and 9 no digit being repeated ?Sol :
the no. of digits = 5
Required no. = ^{5}P_{5} = 5! = 120
Ques 2. How many threedigit numbers can be formed by using the digits in 735621, if repetition is not allowed ?
Sol:
^{n}P_{r} = n! / (nr)!
^{6}P_{3 }= 6! / (63)!
^{6}P_{3 }= 6!/3!
^{6}P_{3 }= 120
Ques 3. Find the number of different words that can be formed from the word 'SUCCESS'.
Sol : No. of Permutation = n! / p! × q!, where p = of one type , q = ( of another type ).
No. of Permutation = 7!/ 3! × 2!
No. of Permutation = 420
Ques 4. How many different 5  digit numbers can be formed by using the digits of the number 713628459 ?
Sol :
^{n}P_{r} = n! / (nr)!
^{9}P_{5} = 9! / (95)!
^{9}P_{5} = 9! / 4!
^{9}P_{5} = 15,120
Ques 5. How many numbers of five digits can be formed with the digits 0,2,4,6 and 8 ?
Sol:
Ques 6. How many numbers of five digits can be formed with the digits 0,1, 2, 3, 4, 6 and 8 ?
Sol :
Here nothing has been said about the repetition of digits. So , it is understood that repetition of digits is not allowed .
Ques 7. How many even numbers of three digits can be formed with the digits 0,1, 2, 3, 4, 5 and 6 ?
Sol :
Total of such numbers = 5 × 5 v 3 = 75
req no. = 30+75 = 105
Ques 8. A round table conference is to be held between delegates of 15 companies. In how many ways can they be seated if delegates from two MNCs may wish to sit together ?
Sol :
Since delegates from two multinational companies will sit together, so considering these two delegates as one unit, there will be 13 + 1 = 14 delegates who can be arranged in a circular table in 14! ways.
The two delegates from the MNCs can be arranged among themselves in 2! ways.
Using the product rule, the required no. of ways = 14!×2!
Ques 9. A person has 12 friends out of which 7 are relatives. In how many ways can he invite 6 friends such that at least 4 of them are relatives ?
Sol:
1. Factorial Notation:
Let n be a positive integer. Then, factorial n, denoted n! is defined as:
n! = n(n – 1)(n – 2) … 3.2.1.
Examples:
2. Permutations:
The different arrangements of a given number of things by taking some or all at a time, are called permutations.
Examples:
3. Number of Permutations:
Number of all permutations of n things, taken r at a time, is given by:
^{n}P_{r} = n(n – 1)(n – 2) … (n – r + 1) =
Examples:
4. An Important Result:
If there are n subjects of which p_{1} are alike of one kind; p_{2} are alike of another kind;p_{3} are alike of third kind and so on and p_{r} are alike of r^{th} kind,
such that (p_{1} + p_{2} + … p_{r}) = n.
Then, number of permutations of these n objects is =
5. Combinations:
Each of the different groups or selections which can be formed by taking some or all of a number of objects is called a combination.
Examples:
AB, AC, AD, BC, BD, CD.
6. Number of Combinations:
The number of all combinations of n things, taken r at a time is:
Note:
Examples:
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