A fair dice is rolled n times. The number of all the possible outcomes is
The sum of all the numbers which can be formed by using the digits 1 , 3 , 5 , 7 ,9 all at a time and which have no digit repeated is
4 boys and 4 girls are to be seated in a row. The number of ways in which this can be done, if the boys and girls sit alternately, is
The letters of the word ‘SOCIETY’ are arranged in such a manner that the vowels and consonants occur alternately, the number of different words so obtained is
There are 10 true-false questions. The number of ways in which they can be answered is
The number of ways in which 6 “ + “ and 4 “ – “ signs can be arranged in a line such that no two “ – “ signs occur together is
′+′ signs can be put in a row in one way creating seven gaps shown as arrows:
Now 4′−′ signs must be kept in these gaps. So, no tow ′−′ signs should be together.
Out of these 7 gaps 4 can be chosen in 7C4 ways.
On a railway track, there are 20 stations. The number of tickets required in order that it may be possible to book a passenger from every station to every other is
Number of tickets selected from first station =20
from second =19
.... for last station =0
We have to select 2 consecutive stations
so total number of possible tickets = P(20,2)
A class is composed 2 brothers and 6 other boys. In how many ways can all the boys be seated at the round table so that the 2 brothers are not seated besides each other?
Take 1 person from 6 and fix him and 5 others can arranged in -- 5! ways=120
there are 6 places left in which 2 brothers can sit
so they can choose any 2 places from 6 - 6C2 ways=15
2 brothers can arrange themselves in 2! ways=15*2=30
The number of all selections which a student can make for answering one or more questions out of 8 given questions in a paper, when each question has an alternative, is:
The number of ways in which 8 different flowers can be strung to form a garland so that 4 particular flowers are never separated is
Different calendars for the month of February are made so as to serve for all the coming years. The number of such calendars is
The mint has to perform two jobs:
1) Selecting the number of days in the February month i.e., 28 or 29
2) Selecting the first day of February month.
The first job can be completed in 2 ways while second job can be completed in 7 ways by selecting any one of the seven days of a week.
Thus, the required number of plates = 2 x 7 = 14.
The number of all odd divisors of 3600 is
Number 1 is odd. As any number is divisible by 1: 1
Do prime factorisation of 3600: 2*2*2*2*3*3*5*5
Select all odd numbers from above: 3,3,5,5
Try every possible products of these:
Single number: 3, 5
Two numbers: 3*3, 3*5, 5*5: 9,15,25
Three numbers: 3*3*5, 3*5*5: 45, 75
All four: 3*3*5*5: 225
The odd divisors are: 1,3,5,9,15,25,45,75,225
There are 9 odd divisors of 3600.
The number of all even divisors of 1600 is
A convex polygon of n sides has n diagonals. The value of n is
The number of all possible positive integral solutions of the equation xyz = 30 is
All possible three number multiplications originate from the following triads:
First one can have 3!/2! = 3 ways and the remaining four triads can have 3! combinations
total combinations = 3 + 4*3! = 27
Number of all 4 digit numbers with distinct digits is
The number of ways, in which a student can choose 5 courses out of 8 courses, when 2 courses are compulsory, is
The number of ways, in which a student can select one or more questions out of 12 each having an alternative, is
No.of choices for each question = 3
There are six questions so total = (3)12
So no.of ways = (3)12 - 1
20 students can compete for a race. The number of ways in which they can win the first three places is (given that no two students finish in the same place)
The number of ways of dividing 52 cards equally into 4 sets is
The number of three digit numbers having atleast one digit as 5 is
These digit number without digit 5 →100....999
→ these are 900 three-digit number
→ from 100 to 199 → 19 number with 5.
total number with 5=19×8+100 for (500-599)
5 boys and 5 girls are to be seated around a table such that boys and girls sit alternately. The number of ways of seating them is
First we fix the alternate position of the girls. Five girls can be seated around the circle in (5−1)!=4! , 5 boys can be seated in five -vacant place by 5!
∴ Required number of ways =4!×5!
The total number of numbers from 1000 to 9999 (both inclusive) that do not have 4 different digits
If P (n,r) = C (n,r) then
126.96.36.199……upto 50 factors is equal to
188.8.131.52... to n factors = (2n)!/n!
184.108.40.206... to 50 factors = (2*2*25)!/25!