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Definition of Fundamental principal of counting
☼ Definition of Fundamental principal of counting : If one operation can be performed in m ways and if corresponding to each of the m ways of performing this operation, there are n ways of performing a second operation, then the number of ways of performing the two operations together is m n.
e.g. (i) Sagar was assigned a roll number for his examination, but he forgot his number. All he remembered was that it was an odd even-digit number whose last digit was neither 2 nor 4. How many numbers fit this description?
Solution.
| Tenth Place | Unit Place |
| Can use : odd no.
1, 3 5, 7, 9 |
Can use : even no.
( ignoring 2, 4 ) |
| 5 ways | 3 ways |
Thus, the total number of ways = 5 3 = 15 ways.
(ii) How many three-digit numbers can be formed from the digits 1, 2, 5 and 7, when no digit is repeated ?
Solution.
| Hundredth Place | Tenth Place | Unit Place |
| Can use : 1, 2, 5, 7 | Can use : remaining 3 digits | Can use : remaining 2 digits |
| 4 ways | 3 ways | 2 ways |
Total no. of three digit nos. using the digits 1, 2, 5, 7 = 4 3
2 = 24
(iii) In how many ways can a selection of two letters be made from the letters X, Y and Z ?
Solution.
| 2nd place | 1st Place |
| Can use : X, Y, Z | Can use : Remaining 2 letters |
| 3 ways | 2 ways |
Thus, there are 3 2 = 6 ways.
They are { (X, Y), (X, Z), (Y, X), (Y, Z), (Z, X), (Z, Y) }
