The letter of the word LABOUR are permuted in all possible ways and the words thus formed are arranged as in a dictionary. What is the rank of the word LABOUR?
The order of each letter in the dictionary is ABLORU.
Now, with A in the beginning, the remaining letters can be permuted in 5! ways.
Similarly, with B in the beginning, the remaining letters can be permuted in 5! ways.
With L in the beginning, the first word will be LABORU, the second will be LABOUR.
Hence, the rank of the word LABOUR is 5!+5!+2 =242
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In how many possible ways can write 3240 as a product of 3 positive integers a,b and c.
We have to distribute three 2's to a, b, c in 3+3−1C3−1 = 5C2 = 10 ways
We have to distribute four 3's to a, b, c in 3+4−1C3−1 = 6C2 = 15 ways
We have to distribute one 5 to a, b, c in 3 ways.
Total ways = 10*15*3 = 450 ways
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In how many ways can the letters of the english alphabet be arranged so that there are seven letter between the letters A and B, and no letter is repeated
We can fix A and B in two ways with 7 letters in between them. Now 7 letters can be selected and arranged in between A and B in 24P7 ways. Now Consider these 9 letters as a string. So now we have 26 - 9 + 1 = 18 letters
These 18 letters are arranged in 18! ways.
So Answer is 2 x 24P7 x 18!
OR 2 x 24P7 x 18! = 36 x 24!.
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How many different integers can be expressed as the sum of three distinct numbers from the set {3, 8, 13, 18, 23, 28, 33, 38, 43, 48}?
A owes B Rs.50. He agrees to pay B over a number of consecutive days starting on a Monday, paying single note of Rs.10 or Rs.20 on each day. In how many different ways can A repay B.
He can pay by all 10 rupee notes in 5 days = 1 way
3 Ten rupee + 1 twenty rupee = 4!/(3!*1!) = 4 ways
1 Ten rupee + 2 twenty rupee notes = 3!/(2!*1!) = 3 ways
Total ways = 1 + 4 + 3 = 8
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A lady has fine gloves and hats in her closet- 14 blue, 20 red, and 18 yellow. The lights are out and it is totally dark. In spite of the darkness, she can make out the difference between a hat and a glove. She takes out an item out of the closet only if she is sure that if it is a glove. How many gloves must she take out to make sure she has a pair of each color?
In this question we are not asked how many ways we can select 3 numbers out of 8.
But how many different numbers can be expressed as a sum of three numbers from the given set.
For example, 3 + 10 + 31 = 3 + 17 + 24 = 47. So 47 can be expressed as a sum of 3 numbers in two different ways but 47 should be considered as only one number.
Now the minimum number that can be expressed as a sum of 3 numbers = 30. (3+10+17)
The next number is 37.
Similarly the largest number is 38 + 45 + 52 = 135.
So there exists many numbers in between, with common difference of 7.
Total numbers = (l?a)/d +1 = (135-30)/7 +1= 15+1=16.
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A college has 10 basketball players. A 5 member team and a captain will be selected out of these 10 players. How many different selections can be made?
We can select the 5 member team out of the 10 in 10C5 ways = 252 ways. The captain can be selected from amongst the remaining 5 players in 5 ways. Therefore, total ways the selection of 5 players and a captain can be made = 252*5 = 1260.
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In how many ways can 5 different toys be packed in 3 identical boxes such that no. box is empty, if any of the boxes may hold all of the toys?
Number of ways of achieving the second option 3 - 1 - 1
Three toys out of the 5 can be selected in 5C3ways. As the boxes are identical, the remaining two toys can go into the two identical looking boxes in only one way.
Therefore, total number of ways of getting the 3 - 1 - 1 option is 5C3 = 10 = 10 ways.
Total ways in which the 5 toys can be packed in 3 identical boxes
= number of ways of achieving Case a + number of ways of achieving Case b
= 15 + 10 = 25 ways.
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How many 4-bit digit numbers that do not contain the digits 3 or 6 are there?
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