Question 33. How many four digit natural numbers not exceeding 4321 can be formed with the digits 1,2,3,4, if the digits can repeat?
Solution:
Total numbers formed by these digits = 4 x 4 x 4 x 4 = 256 (At all 4 positions, 4 options)
Numbers exceeding 4321 are
At first position 4 is fixed
Case 1: 3 at second position
3 at 3rd digit = 4 numbers
4 at 3rd digit = 4 numbers
2 at 3rd and more than digit 1 at 4th = 3 numbers possible
Case 2: 4 at second position = 4 x 4 = 16 numbers (since 4 possibilities for 3rd as well as the 4th digits)
Such numbers = Case 1 + Case 2 = 4 + 4 + 3 + 16 = 27
Total required such numbers = 256 – 27 = 229
Question 34. How many numbers of six digits can be formed from the digits 0,1,3,5,7 and 9 when no digit is repeated? How many of them are divisible by 10?
Solution:
First digit = 5 options (except 0)
Remaining = 5! (since 5 positions, 5 digits)
Total numbers = 5 x 5! = 600
Divisible by 10 = 0 at 6th position, remaining 5 digits and 5 positions – 5! = 120 numbers
Question 35. If three six faced die each marked with numbers 1 to 6 on six faces, are thrown find the total number of possible outcomes.
Solution:
For each dice, number of possible outcomes = 6
Three dice = 6 x 6 x 6 = 216 possibilities
Question 36. A coin is tossed three times and the outcomes are recorded. How many possible outcomes are there? How many possible outcomes if the coin is tossed four times? Five times? n times?
Solution:
At each toss, 2 different possible outcomes
3 times coin tossed = 2 x 2 x 2 = 23 = 8
4 times = 2 x 2 x 2 x 2 = 24 = 16
5 times = 2 x 2 x 2 x 2 x 2 = 25 = 32
n times = 2n
Question 37. How many numbers of four digits can be formed with the digits 1,2,3,4,5 if the digits can be repeated in the same number?
Solution:
For each digit = 5 possible numbers
So, since 4 digits = 5 x 5 x 5 x 5 = 625 numbers
Question 38. How many three-digit numbers can be formed by using the digits 0,1,3,5,7 while each digit may be repeated any number of times?
Solution:
For 1st digit = 4 possible
For 2nd & 3rd = 5 possible
So, 4 x 5 x 5 = 100 such numbers
Question 39. How many natural numbers less than 1000 can be formed from the digits 0,1,2,3,4,5 when a digit may be repeated any number of times?
Solution:
Given: Total number = 6
As we know that the natural ten’s than 1000 can be 1, 2 and 3 digit numbers
So, 0 cannot be the first digit of the 3-digit number.
Now, the hundred’s place can be filled with any of the 5 digits = 5 ways
The ten’s place can be filled with any of the 6 digits = 6 ways
The unit place can be filled with any of the 6 digits = 6 ways
Therefore, the total 3 digit number = 5 x 6 x 6 = 180
The total 2 digit number = 5 x 6 = 30
The total 1 digit number = 5
So, 180 + 30 +1 = 215 natural numbers.
Question 40. How many five 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?
Solution:
Need to select digits for 3rd, 4th and 5th positions
So, 8 options for 3rd position and 7 for 4th position and 6 for 5th one
Total numbers are = 8 x 7 x 6 = 336
Question 41. Find the number of ways in which 8 distinct toys can be distributed among 5 children.
Solution:
Each toy has 5 options.
So, 8 toys = 58 = 390625
Question 42. Find the number of ways in which one can post 5 letters in 7 letter boxes.
Solution:
Total number of letters = 5
Total number of letter box = 7
So, each letter has 7 options and 5 total letters = 75 = 16807
Question 43. Three dice are rolled. Find the number of possible outcomes in which at least one dice shows 5.
Solution:
Total number of outcomes = 6 x 6 x 6 = 216
Outcomes having no 5 = 5 x 5 x 5 = 125
Required outcomes = 216 – 125 = 91
Question 44. Find the total number of ways in which 20 balls can be put into 5 boxes so that first box contains just one ball.
Solution:
First box = 20 options
Other 19 balls have = 4 options = 419 arrangements for 4 boxes
Total ways = 20 x 419
Question 45. In how many ways can 5 different balls be distributed among three boxes?
Solution:
Total number of balls = 5
Total number of boxes =3
So, each ball has 3 options for the boxes = 35 = 243
Question 46. In how many ways can 7 letters be posted in 4 letter boxes?
Solution:
Total number of letters = 7
Total number of letter box = 4
So, each letter has 4 options and 7 total letters = 47 = 16384
Question 47. In how many ways can 4 prizes be distributed among 5 students, when
(i) No students gets more than one prize?
(ii) A student may get any number of prizes?
(iii) No student gets all the prizes?
Solution:
(i) So, 1 student does not get a prize
Choose that student among 5 students 5 different possible ways
Distribute among remaining students = 4! ways = 24
Total ways = 5 x 24 = 120
(ii) Every prize has 5 options
4 prizes = 5 x 5 x 5 x 5 = 625
(iii) Subtract the case of a student to get all prizes from (ii)
Selecting a student to give all prize = 5 ways
= 625 – 5 = 620
Question 48. There are 10 lamps in a hall. Each one of them can be switched on independently. Find the number of ways in which the hall can be illuminated.
Solution:
Each lamp has 2 possibilities = on/off
Total ways = 210 = 1024
One way, all off = need to subtract it
1024 – 1 = 1023 ways