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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?
The bag contains 2 White, 3 Black and 4 Red balls.
So, total 9 balls are there in the bag; among them 3 are Black and 6 are nonBlack balls.
Three balls can randomly be drawn in (9C3) = 84 ways.
1 Black and 2 nonBlack balls can be drawn in (3C1)*(6C2) = 45 ways.
1 nonBlack and 2 Black balls can be drawn in (6C1)*(3C2) = 18 ways.
3 Black balls can be drawn in (3C3) = 1 way.
So, three balls drawn in (45 + 18 + 1) = 64 ways will have at least one Black ball among the drawn ones.
We know , ^{n}P_{r} = n ! /( n  r) !
So ^{100}P_{2 }= 100 ! / ( 100  2 ) !
= 100 ! / 98 !
= 100 x 99 x 98 ! / 98 !
= 9900
So Option D is correct answer.
A coin is tossed 3 times. Find out the number of possible outcomes.
For any multiple independent event, there are n^{m}
total possible outcomes, where n is the number of outcomes per event, and m is the number of such events.
So for a coin, discounting the unlikely event of landing on its side, there are two possible outcomes per event, heads or tails. And it is stated that there are 3 such events. So n^{m}=2^{3}=8
.
In how many ways can the letters of the word 'LEADER' be arranged?
The word 'LEADER' has 6 letters.
But in these 6 letters, 'E' occurs 2 times and rest of the letters are different.
Hence,number of ways to arrange these letters
How many 3 digit numbers can be formed from the digits 2, 3, 5, 6, 7 and 9 which are divisible by 5 and none of the digits is repeated?
Since each desired number is divisible by 5, so we must have 5 at the unit place. So, there is 1 way of doing it.
The tens place can now be filled by any of the remaining 5 digits (2, 3, 6, 7, 9). So, there are 5 ways of filling the tens place.
The hundreds place can now be filled by any of the remaining 4 digits. So, there are 4 ways of filling it.
∴ Required number of numbers = (1 x 5 x 4) = 20.
In how many different ways can the letters of the word 'JUDGE' be arranged such that the vowels always come together?
The given word contains 5 different letters.
Keeping the vowels UE together, we suppose them as 1 letter.
Then, we have to arrange the letters JDG (UE).
Now, we have to arrange in 4! = 24 ways.
The vowels (UE) can be arranged among themselves in 2 ways.
∴ Required number of ways = (24 × 2) = 48
How many words with or without meaning, can be formed by using all the letters of the word, 'DELHI' using each letter exactly once?
Explanation :
The word 'DELHI' has 5 letters and all these letters are different.
Total words (with or without meaning) formed by using all these
5 letters using each letter exactly once
= Number of arrangements of 5 letters taken all at a time
= ^{5}P_{5} = 5! = 5 x 4 x 3 x 2 x 1 = 120
How many arrangements can be made out of the letters of the word 'ENGINEERING' ?
The number of arrangements of the word ENGINEERING is 277200.
Solution:
ENGINEERING word has 3 times of 3, three times of N, 2 times of G and 2 times of I. Then, the total letter is 11.
So, the number of arrangements of the word ENGINEERING = 11!/[3! * 3! * 2! * 2!]
= 39916800/[6 * 6 * 2 * 2]
= 277200
Hence, the number of arrangements of the word ENGINEERING is 277200.
How many words can be formed by using all letters of the word 'BIHAR'?
Explanation :
The word 'BIHAR' has 5 letters and all these 5 letters are different.
Total words formed by using all these 5 letters = ^{5}P_{5} = 5!
= 5 x 4 x 3 x 2 x 1 = 120
In how many different ways can the letters of the word 'DETAIL' be arranged such that the vowels must occupy only the odd positions?
There are 6 letters in the given word, out of which there are 3 vowels and 3 consonants.
Let us mark these positions as under:
(1) (2) (3) (4) (5) (6)
Now, 3 vowels can be placed at any of the three places out 4, marked 1, 3, 5.
Number of ways of arranging the vowels = 3P3
= 3! = 6.
Also, the 3 consonants can be arranged at the remaining 3 positions.
Number of ways of these arrangements = 3P3
= 3! = 6.
Total number of ways = (6 x 6) = 36.
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