Where Mindful Learning Begins
Where Mindful Learning Begins
Free, step-by-step Class 11 Maths NCERT Solutions for Chapter 6 Ex 6.1 — all 6 questions solved, covering the Fundamental Principle of Counting: forming numbers, codes and signals, with and without repetition allowed.
Every question in this exercise comes back to one idea: if one stage of a task can be done in m ways, and a following stage in n ways, the whole task can be done in m\times n ways. The only real skill being tested is filling vacant places, one at a time, in the correct order — and being careful about whether a choice can repeat, and whether any position (like a leading digit, or a units digit that must be even) has a restriction the others don't.
A 3-digit number has three places to fill — hundreds, tens and units — each from the 5 given digits.
Each of the 3 places can be filled in 5 ways, since a digit used once can be used again.
5\times5\times5=125
The hundreds place can be filled in 5 ways. Once that digit is used, the tens place can be filled in only 4 ways (one digit is gone), and the units place in 3 ways.
5\times4\times3=60
A 3-digit number is even exactly when its units digit is even, so the units place must be filled first, from the restricted choices.
Units place: only 2, 4 or 6 can go here, so there are 3 ways.
Tens place: any of the 6 digits can go here (repetition allowed), so there are 6 ways.
Hundreds place: any of the 6 digits can go here as well, so there are 6 ways.
By the multiplication principle, the required number of 3-digit even numbers is:
6\times6\times3=108
There are 4 vacant places to fill, one for each letter of the code, using the first 10 letters of the alphabet (A to J), with no letter repeated.
The 1st place can be filled in 10 ways. Since repetition is not allowed, the 2nd place can be filled in 9 ways, the 3rd place in 8 ways, and the 4th place in 7 ways.
By the multiplication principle, the required number of 4-letter codes is:
10\times9\times8\times7=5040
The first two digits of every telephone number are fixed as 6 and 7, so only the remaining 3 places need to be filled, using the 8 digits left over (0–9 excluding 6 and 7), with no digit repeated.
The 3rd place can be filled in 8 ways. The 4th place can then be filled in 7 ways, and the 5th place in 6 ways.
By the multiplication principle, the required number of telephone numbers is:
8\times7\times6=336
Each toss of the coin can result in either Head or Tail, so each toss has 2 possible outcomes, independent of the other tosses.
By the multiplication principle, for 3 tosses in succession, the total number of possible outcomes is:
2\times2\times2=2^3=8
A signal is made by filling 2 vacant places, one below the other, using flags from the 5 available flags of different colours, with no flag repeated in the same signal.
The upper place can be filled by any one of the 5 flags, in 5 ways. Once that flag is used, the lower place can be filled by any of the remaining 4 flags, in 4 ways.
By the multiplication principle, the required number of signals is:
5\times4=20
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