Exercise 6.1
Permutations and combinations
Question and Answers
Class 11 – Maths
| Class | Class 11 |
| Subject | Mathematics |
| Chapter Name | Permutations and Combinations |
| Chapter No. | Chapter 6 |
| Exercise | Exercise 6.1 |
| Category | Class 11 Maths NCERT Solutions |
Question 1 How many 3-digit numbers can be formed from the digits 1, 2, 3, 4 and 5 assuming that
(i) repetition of the digits is allowed?
Answer
Let the 3-digit number be XYZ, where Z is at the units place, Y at the tens place and X at the hundreds place. When repetition is allowed, the units place can be filled in by any of the all 5 digits. Similarly, tens and hundreds digits can be filled by any of the all 5 digits. Thus, the number of ways in which 3-digit numbers can be formed from the given digits is 5×5×5 = 125.
(ii) repetition of the digits is not allowed?
Answer
Let the 3-digit number be XYZ, where Z is at the units place, Y at the tens place and X at the hundreds place. When repetition is not allowed, if units place is filled at first, then it can be filled by any of the given 5 digits. Thus, the number of ways is 5. Similarly, the tens place can be filled by remaining 4 digits and the hundreds place can be filled by remaining three digits. Therefore, by multiplication principle, the number of ways = 5×4×3 = 60.
Question 2 How many 3-digit even numbers can be formed from the digits 1, 2, 3, 4, 5, 6 if the digits can be repeated?
Answer Let the 3-digit number be XYZ, where Z is at the units place, Y at the tens place and X at the hundreds place. The units place can be filled in by any of the all 6 digits. Similarly, tens and hundreds digits can be filled by any of the all 6 digits, as repetition is allowed. Therefore, The total number of possible 3-digit numbers = 6 × 6 × 3 = 108
Question 3 How many 4-letter code can be formed using the first 10 letters of the English alphabet, if no letter can be repeated?
Answer As repetition is not allowed, the first place can be filled in 10 different ways by any of the given 10 letters and second place can be filled in 9 different ways. Similarly third place can be filled in 8 different ways and the fourth in 7 different ways.
Therefore, the total number of 4-letter codes=10 × 9 × 8 × 7 = 5040
Question 4 How many 5-digit telephone numbers can be constructed using the digits 0 to 9 if each number starts with 67 and no digit appears more than once?
Answer Given that 5 digit telephone number starts with 67. Let the five-digit number be PQRST. Therefore, the number is 67RST. As repetition is not allowed and 6 and 7 are already taken, the digits available for place C are 0,1,2,3,4,5,8,9. Thus, units place can be filled in 8 different ways, tens place can be filled in 7 different ways and hundreds place can be filled in 6 different ways.
∴ The total five-digit numbers with given conditions = 8 × 7 × 6 = 336
Question 5 A coin is tossed 3 times and the outcomes are recorded. How many possible outcomes are there?
Answer The possible outcomes after a coin toss are head and tail i.e., the number of ways of showing different face = 2
∴ The total number of possible outcomes after 3 times = 2 × 2 × 2 = 8
Question 6 Given 5 flags of different colours, how many different signals can be generated if each signal requires the use of 2 flags, one below the other?
Answer The signal requires 2 flags. Given 5 flags of different colours.
The upper empty place can be filled in 5 different ways and the lower empty place can be filled in 4 different ways.
The number of ways in which signal can be given = 5 × 4 = 20
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