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Out of 7 consonants and 4 vowels, how many words of 3 consonants and 2 vowels can be formed ?
No. of ways of selecting 3 consonants from 7 .
No. of ways of selecting 2 vowels from 4 .
No. of ways of selecting 3 consonants from 7 and 2 vowels from 4
No. of ways of arranging 5 letters among themselves
In a group of 6 boys and 4 girls, foure children are to be selected. In how many different ways can they be selected st at least one boy should be there ?
According to the given data we have 4 options.
Opt.1 :
We can select 4 boys from 6
Opt.2 :
We can select 3 boys and one girl
Opt.3 :
We can select 2 boys and 2 girls
Opt.4 :
We can select 1 boy and 3 girls
From a group of 7 men and 6 women, five persons are to be selected to form a group so that at least 3 men are there in this group. In how many ways it can be done ?
According to above data we have 3 options.
Opt.1 :
We can select 5 men .
Opt.2 :
We can select 4 men and 1 women .
Opt.3 :
We can select 3 men and 2 women .
In how many different ways can the letters of the word ‘OPTICAL’ be arranged so that the vowels always come together ?
The word ‘OPTICAL’ has 7 letters and 3 vowels as vowels always come together we consider the words will be 5 letters such like PTLC(OIA).
Hnece the no. of ways arrange these letters
Number of ways arrange these vowels among themselves
In how many ways can the letters of the word ‘CORPURATION’ be arranged so that the vowels always come together ?
The word ‘CORPURATION’ has 11 letters and it has 5 vowels which always come together and formed as a single letter.
i.e CRPRTN(OOAIO).
Hence we can assume 7 letters, but here ‘R’ accurs 2 times.
In the 5 vowels, ‘O’ occurs 3 times
In how many ways can a group of 5 men and 2 women be made out of a total of 7 men and 3 women ?
We need to select 5 men from 7 and 2 women from 3.
In how many different ways can the letters of the word ‘MATHEMATICS’ be arranged st the vowels always come together ?
‘MATHEMATICS’ has 11 letters and 4 vowels. These 4 vowels can be grouped and consider as a single letter, i.e MTHMTCS(AEAI).
Hence we assume total letters as 8. But here ‘M’ acurs 2 times, ‘T’ accurs 2 times, hence the no. of ways to arrange
In 4 vowels ‘A’ accurs 2 times, no. of ways to arrange among themeselves
There are 8 men and 10 women and you need to form a committe of 5 men and 6 women. In how many ways can the committe be formed ?
We need to select 5 men from 8 and 6 women from 10.
How many 3letters words can be formed out of the letters of the word ‘LOGIC’ if the repetition of letters are not allowed ?
‘LOGIC’ has 5 different letters. Total no. of 3letters words formed
In how many different ways can the letters of word ‘LEADING’ be arranged st the vowels should always come together ?
‘LEADING’ has 7 letters and 3 vowels must always come together, i.e LDNG(AEI).
Hence we can assume total 5 letters.
No. of ways arrange these letters .
No. of ways arrange the vowels .
.
A coin is tossed 3 times. Find out the number of possible outcomes.
When a coin is tossed, there are two possible outcomes, i.e H,T. When a coin is tossed 3 times, then the no. of possible outcomes
In how many different ways can the letters of the word ‘DETAIL’ be arranged s.t the vowels must occupy only the odd positions ?
‘DETAIL’ has 6 letters, (3 vowels, 3 consonants). The vowels must occupy only the odd positions i.e in (1), (3) and (5).
No. of ways to arrange these vowels
No. of cosonants which is arranged in remaining 3 positions in any order
A bag contains 2 white balls, 3 black balls and 4 red balls. In how many ways can 3 balls be drawn from the bag, if at least one black ball is to be included in the draw ?
We have 3 options.
Opt 1: We can select 3 black balls .
Opt 2 : 2 black, 1 nonblack .
Opt 3 : 1 black, 2 nonblack .
In how many ways can the letters of the word ‘JUDGE’ be arranged st the vowels always come together ?
‘JUDGE’ has 5 letters (2 vowels). We can assume total letters as 4 i.e JDG(UE).
No. of ways to arrange these letters .
No. of ways to arrange these vowels among themselves .
.
In how many ways can the letters of the word ‘LEADER’ be arranged ?
‘LEADER’ has 6 letters.
‘E’ occurs 2 times.
No. of ways to arrange these letters .
How many words can be formed by using all letters of the word ‘ODISHA’ ?
‘ODISHA’ has 6 different letters.
Total no. of words can be formed by using 6 letters .
How many arrangements can be made out of the letters of the word ‘ENGINEERING’ ?
‘ENGINEERING’ has 11 letters.
‘E’ occurs 3 times .
‘N’ occurs 3 times.
‘G’ occurs 2 times.
‘I’ occurs 2 times.
How many 3digit numbers can be formed from the digits 2,3,5,6,7 and 9 which are divisible by 5 and no repeatation allow ?
A number is divisible by 5 if its unit place contains 0 or 5.
From the above no. we have only 5, there is only 1 way to select 5 in unit place.
We have 5 no.s left.
Hence 10^{th} place contains 5 elements
Since no repeatation we have 4 elements to be placed at 100^{th} place.
How many words can be formed by using the letters of the word, ‘DELHI’ using each letter exactly once.
‘DELHI’ has 5 different letters.
No. of arrangements of 5 letters taken all at a time
What is the value of \(^{100}P_{2} \) ?
In how many different ways can the letters of the word ‘RUMOUR’ be arranged ?
‘RUMOUR’ has 6 letters.
‘R’ occurs 2 times.
‘U’ occurs 2 times.
.
There are 6 periods in each working day of a school. In how many ways can one organize 5 subjects st each subject is allowed at least one period ?
5 subjects can be selected in ways 1 subject can be selected in ways, these 6 subjects can be arranged themselves in ways.
Repeatation of at least one subject is allowed from 5 subjects to select 6 periods.
How many 6 digit telephone numbers can be formed if each number starts with 35 and no digit appears more than once ?
The two places can only be filled by 3 and 5 respectively, there is only 1 way.
Given that no of digit appears more than once, hence we have 8 digits remaining, (0,1,2,4,6,7,8,9).
Then 4 places can be filled by ways.
An event manager has ten patterns of chairs and eight patterns of tables. In how many ways can he make a pair of table and chair ?
He has 10 patterns of chairs and 8 patterns of tables.
A chair can be selected in 10 ways.
A table can be selected in 8 ways.
ways.
25 buses are running between P and Q places. In how many ways can a person go from P to Q and return by different bus ?
There are 25 buses.
He can go in any of the 25 buses, i.e 25 ways.
But he can’t came in same bus.
Hence he had 24 ways.
A box contain 4 red, 3 white and 2 blue balls. Three balls are drawn at random. Find out the number of ways of selecting the balls of different colours ?
Red ball can be selected in ways.
White ball can be selected in ways.
Blue ball can be selected in ways.
A question paper has two parts P and Q, each containing 10 questions. If a student needs to choose 8 form par part P and 4 from part Q, in how many ways can he/she do that ?
No. of ways to choose 8 questions from part P
No. of ways to choose 4 questions from part 4 questions from part Q
In how many ways can 5 girls and 5 boys form a circle st the boys and the girls alternate each other ?
Around a cercle, 5 boys can be arranged in ways.
Given that the boys and the girls alternate each other. Hence there are 5 places for the girls.
Therefore the girls can be arranged in ways.
Total no. of ways
Find out the number of ways in which 6 rings of different types can be worn in 3 fingers ?
Each ring can be worn in 3 ways there are 6 rings.
Total no of ways
In how many ways can 5 man draw water from 5 taps if no tap can be used more than once ?
1^{st} man can draw water from any of the 5 taps.
Similarly, 2^{nd}, 3^{nd}, 4^{th}, 5^{th} men can draw water from 4,3,2,1 ways respectively.