Question 1: In how many ways can the letters of the word ABACUS be rearranged such that the vowels always appear together?
Question 2: How many different four letter words can be formed (the words need not be meaningful) using the letters of the word MEDITERRANEAN such that the first letter is E and the last letter is R?
Question 3:What is the probability that the position in which the consonants appear remain unchanged when the letters of the word "Math" are re-arranged?
Question 4: There are 6 boxes numbered 1, 2, ... 6. Each box is to be filled up either with a red or a green ball in such a way that at least 1 box contains a green ball and the boxes containing green balls are consecutively numbered. The total number of ways in which this can be done is:
Question 5: A man can hit a target once in 4 shots. If he fires 4 shots in succession, what is the probability that he will hit his target?
Question 6: In how many ways can 5 letters be posted in 3 post boxes, if any number of letters can be posted in all of the three post boxes?
Question 7: Ten coins are tossed simultaneously. In how many of the outcomes will the third coin turn up a head?
Question 8: In how many ways can the letters of the word "PROBLEM" be rearranged to make seven letter words such that none of the letters repeat?
Saturday, November 20, 2010
Friday, November 19, 2010
Combinations
Combinations
Combinations are of two types:
1 In which repetition is allowed
2. In which repetion is not allowed
Combinations with repetition
(n+r-1)!/r!(n-1)!
Where n is the number of things to choose from and you choose r of them
Combinations without repetition
A simple example of this would be lottery numbers.
If we assume that the order does matter, i.e., permutations, alter it so the order does not matter.
Formula: n!/r!(n-r)!
Where n is the number of things to choose from, and you choose r of them
It is also known as "n choose r" or binomial coefficient.
Combinations are of two types:
1 In which repetition is allowed
2. In which repetion is not allowed
Combinations with repetition
(n+r-1)!/r!(n-1)!
Where n is the number of things to choose from and you choose r of them
Combinations without repetition
A simple example of this would be lottery numbers.
If we assume that the order does matter, i.e., permutations, alter it so the order does not matter.
Formula: n!/r!(n-r)!
Where n is the number of things to choose from, and you choose r of them
It is also known as "n choose r" or binomial coefficient.
Thursday, November 18, 2010
Permutations
Permutations
Permutation is the technique of permuting or rearranging objects or values in a particular order.
In permutations the order matters, however, in combination the order does not matter.
Permutations are of two types:
1. Where repetition is allowed
2. Where no repetition is allowed
How to calculate permutation with repetition?
If you have n things and n choices each time. And you have to choose r of them:
The Permutations are as follows: n x n x ........(r times) or n^r
How to calculate permutation without repetition?
In this you have to reduce the number of available choices each time.
Formula: n!/(n-r)!
Where n is the number of things to choose from, and you choose r of them.
Permutation is the technique of permuting or rearranging objects or values in a particular order.
In permutations the order matters, however, in combination the order does not matter.
Permutations are of two types:
1. Where repetition is allowed
2. Where no repetition is allowed
How to calculate permutation with repetition?
If you have n things and n choices each time. And you have to choose r of them:
The Permutations are as follows: n x n x ........(r times) or n^r
How to calculate permutation without repetition?
In this you have to reduce the number of available choices each time.
Formula: n!/(n-r)!
Where n is the number of things to choose from, and you choose r of them.
Tuesday, November 16, 2010
Arithmetic and Geometric Progressions
Arithmetic Progressions
These are series of numbers e.g. 2, 6, 10, 14, 18 ... or 100, 97.5, 95, 92.5, 90, . . . . where a constant number is added to or subtracted from each term to form the next.
Assuming that the first term is a and that the number added/subtracted (the "common difference") is d, then the series is: a, a + d, a + 2d, a + 3d .... The nth term of the series is a + (n - 1)d.
The sum of the first n terms of the series is: 0.5n(2a + [n - 1]d).
Geometric Progressions
These are series of numbers, e.g. 2, 6, 18, 54, 162, ... or 100, -50, 25, -12.5, 6.25 ... where each term is multiplied/divided by a constant number (the "common ratio") to form the next. The number can be positive (in which case all the terms are the same sign) or negative (in which case alternate terms have alternate signs). The first term is usually called a. The common ratio is usually called r.
If r is outside the range -1 to 1, the terms of the series get bigger and bigger (even if they change sign), and the series diverges. If r is within the range -1 to 1, the terms get smaller and smaller (closer to 0) and the series converges.
The sum of the first n terms of a geometric progression is a(1 - r n)/1 - r
If r lies within the range -1 < r < 1, then the series has a sum to infinity (i.e. if you added up an infinite number of terms of the series, you would still get a finite number). An example is the series 1, 1/2, 1/4, 1/8, 1/16 ... etc. where a = 1 and r = 1/2. In this case, if you add the terms of the series, you get closer and closer to 2 without ever reaching it. If you could add up an infinite number of terms, then the sum would be 2.
To find the sum to infinity of the series, simply put n = ¥ in the equation above. Providing -1 < r < 1, the term r¥ will be 0, and the whole formula reduces to this:
a/1 - r
These are series of numbers e.g. 2, 6, 10, 14, 18 ... or 100, 97.5, 95, 92.5, 90, . . . . where a constant number is added to or subtracted from each term to form the next.
Assuming that the first term is a and that the number added/subtracted (the "common difference") is d, then the series is: a, a + d, a + 2d, a + 3d .... The nth term of the series is a + (n - 1)d.
The sum of the first n terms of the series is: 0.5n(2a + [n - 1]d).
Geometric Progressions
These are series of numbers, e.g. 2, 6, 18, 54, 162, ... or 100, -50, 25, -12.5, 6.25 ... where each term is multiplied/divided by a constant number (the "common ratio") to form the next. The number can be positive (in which case all the terms are the same sign) or negative (in which case alternate terms have alternate signs). The first term is usually called a. The common ratio is usually called r.
If r is outside the range -1 to 1, the terms of the series get bigger and bigger (even if they change sign), and the series diverges. If r is within the range -1 to 1, the terms get smaller and smaller (closer to 0) and the series converges.
The sum of the first n terms of a geometric progression is a(1 - r n)/1 - r
If r lies within the range -1 < r < 1, then the series has a sum to infinity (i.e. if you added up an infinite number of terms of the series, you would still get a finite number). An example is the series 1, 1/2, 1/4, 1/8, 1/16 ... etc. where a = 1 and r = 1/2. In this case, if you add the terms of the series, you get closer and closer to 2 without ever reaching it. If you could add up an infinite number of terms, then the sum would be 2.
To find the sum to infinity of the series, simply put n = ¥ in the equation above. Providing -1 < r < 1, the term r¥ will be 0, and the whole formula reduces to this:
a/1 - r
Monday, November 15, 2010
Number Series
How to solve number series questions?
There are different ways and relations to solve the number series questions, some of them are as follows:
1. Difference
2. Product
3. Squares/Cubes
4. Prime/Composite
5. Combination
6. Counting
7. Miscellaneous
There are different ways and relations to solve the number series questions, some of them are as follows:
1. Difference
2. Product
3. Squares/Cubes
4. Prime/Composite
5. Combination
6. Counting
7. Miscellaneous
Thursday, November 11, 2010
What is Mode?
Mode is the number that appears most often in a list.
For example: In a series: 2, 4, 5, 6, 8, 5, 6, 7, 9, 2, 1, 5
The mode is 5 as it appears three times.
For example: In a series: 2, 4, 5, 6, 8, 5, 6, 7, 9, 2, 1, 5
The mode is 5 as it appears three times.
Wednesday, November 10, 2010
GMAT Practice Questions
1- The smallest number which, when divided by 4,6,or 7 leaves a remainder of 2, is:
2- If ROPE is coded as 6821 & CHAIR is coded as 73456 then what will be the code for CRAPE?
3- What is the smallest number which when increased by 5 is completely
divisible by 8,11, and 24?
GMAT Practice Questions
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