Question: How many 4-letter words can be formed using the alphabets of the word ENGLISH, if it is given that the 4-letter word contains alphabets G and L and repetition of alphabets are not allowed?

1. 60
2. 120
3. 180
4. 200
5. 240

Answer: E
Solution and Explanation:

Approach Solution 1:
Given that 4 letter words should contain the alphabets G and L and repetition is not allowed. We have to count how many words can be formed.
Here, we have 7 letters ( E, N,G, L, I, S & H )
in 4 locations as follows:
G and L must be present in order to arrange the first two letters in two different ways, giving us 120 possible arrangements for the remaining five letters (E, N, I, S, and H).
Therefore, there are 240 ways in total (120 * 2)
E is the correct answer.

Approach Solution 2:
First task: Choose two letters from the groups G and L. (Since these two must be included, obviously)
Choose two letters from the list: E, N, I, S, and H.

Task 3: Create 4-letter words using the four letters you chose for the first two challenges.
The first task is to choose two letters from the groups G and L.
Now, there are 2C2 = 1 methods to choose 2 letters from 2 different letters.
Thus, there is only one way to complete Task 1.

Task 2 asks you to choose two letters from the set of five letters (EN, I, S, H).
There are 5C2 = 10 methods to choose 2 letters from a pool of 5 different letters.
Thus, there are 10 different ways to complete Task 2.

Task 3 asks you to arrange the four letters you've chosen in four spaces to create various words.
The number of possible arrangements for 4 letters in 4 spaces is 4! = 4 X 3 X 2 X 1.
Thus, there are 24 different methods to complete Task 3.

Step 4: Determine the final response.
We will enter the following values into the aforementioned equation in this step:
There are 240 different 3-letter words, or 1 x 10 x 24.
As a result, 240 words can be produced within the circumstances mentioned in the question.
E is the correct choice.

Approach Solution 3:

Stage 1: Choose two letters from the list E, N, I, S, or H.
We can utilise combinations because it doesn't matter what order we choose the two letters in (yet!).
There are 5C2 methods to choose 2 letters, 5 letters (10 ways)
Therefore, there are 10 ways we can finish stage 1.
Aside: Below is a video on how to mentally calculate combinations (like 5C2) if you're interested.

Stage 2: Combine the letters G and L with the two that you selected in stage 1, then arrange those four letters.
We have n ways to arrange the things.
As a result, there are 24 possible arrangements for the four letters.
We have 24 ways to complete stage 2.
By using the Fundamental Counting Principle (FCP), there are 240 possible methods to finish the two steps and produce four-letter words.
E is the correct choice.

“How many 4-letter words can be formed using the alphabets of the word" - is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been borrowed from the book “GMAT Official Guide Quantitative Review”.

To understand GMAT Problem Solving questions, Candidates must have basic qualitative abilities. Quant evaluates a candidate's aptitude for both mathematics and logic. The problem-solving section of the GMAT Quantitative test consists of a question and a list of potential answers. The candidate must choose the right answer by applying maths to the question. The problem-solving section of the GMAT Quant topic is made up of very complicated maths problems that must be solved by using the right maths facts.

Suggested GMAT Problem Solving:

• After 6 Games, Team B Had an Average of 61.5 Points Per Game. GMAT Problem Solving
• If 12 Ounces of a Strong Vinegar Solution are Diluted with 50 Ounces of Water to Form a Three-Percent Vinegar Solution GMAT Problem Solving
• A Contractor Estimated that his 10-Man Crew Could Complete the Construction in 110 Days if There Was no Rain. GMAT Problem Solving
• At a Dog Competition, a Dog is Awarded 10 Points if it Runs Through 4 Pipes, Makes 10 Jumps, and Walks on 2 Beams. GMAT Problem Solving
• The Population of the Bacteria Colony Doubles Every Day GMAT Problem Solving
• If tu=x tu=x and y=uxy=ux Where t, u, x, and y are Non-Zero Integers GMAT Problem Solving
• The farm has Chickens, Cows and Sheep. The Number of Chickens and Cows Combined is 3 Times the Number of Sheep. GMAT Problem Solving
• In how Many Different Ways Can a Group of 8 People be Divided into 4 Teams of 2 People Each? GMAT Problem Solving
• If m is Three Times n, and if 2n + 3 is 20% of 25, What is the value of m? GMAT Problem Solving
• If Ben Were to Lose the Championship, Mike would be the Winner GMAT Problem Solving
• A Train Travelling at a Certain Constant Speed takes 30 seconds GMAT Problem Solving
• A Conference Room is Equipped with a Total of 45 Metal or Wooden Chairs.GMAT Problem Solving
• How many litres of a 90% solution of concentrated acid needs to be mixed with a 75% solution GMAT problem solving
• Jug Contains Water And Orange Juice In The Ratio 5:7 . Another Jug Contains Water And Orange J GMAT problem solving
• The number of ways in which 8 different flowers can be seated to form a garland so that 4 particular flowers are never separated GMAT problem solving
• A train travels from Albany to Syracuse, a distance of 120 miles, at an average rate of 50 miles per hour GMAT problem solving
• The hexagon ABCDEF is regular. That means all its sides are the same length and all its interior angles are the same size. GMAT problem solving
• y varies directly as x and when x = 6, y = 24. What is the value of y, when x = 5? GMAT problem solving
• If k is an Integer and 2 < k < 7, for How Many Different Values of k is There a Triangle With Sides of Lengths 2, 7, and k? GMAT problem solving
• How many factors does 36^2 have? GMAT problem solving
• A number when divided successively by 4 and 5 leaves remainder 1 and 4 respectively GMAT problem solving
• Walking at 6/7 th of his usual speed, a man is 25 min too late GMAT problem solving
• An Inlet Pipe can Fill in an Empty Cistern in 30 minutes Whereas a leak in the Bottom of the Cistern can Empty a Filled Tank in 40 minutes GMAT problem solving
• A milkman cheats his customers by adding water to the milk he sells GMAT problem solving
• if 80 lamps can be lighted, 5 hours per day for 10 days for $21.25, then the number of lamps, GMAT problem solving
• Few of the corporate contributions to the earthquake relief fund, aside from Pterocom GMAT problem solving
• The product of the first 10 prime numbers is closest to which of the following? GMAT problem solving
• There is a 120 liter mixture of alcohol and water. The ratio of alcohol to water is 7 : 5 GMAT problem solving
• An equilateral triangle ABC is inscribed in square ADEF, forming three right triangles GMAT problem solving
• A Furniture Store Sells Only Two Models of Desks, Model A and Model B. The Selling Price of Model A is $120 GMAT problem solving
• A contractor combined x tons of a gravel mixture that contained 10 percent gravel G GMAT problem solving
• There are 100 Apples in a Bag of which 98% are Green and Rest are Red GMAT problem solving
• The Smallest of Six Consecutive Odd Integers Whose Average (arithmetic mean) is x + 2 GMAT Problem Solving
• Two Consultants, Mary and Jim, Can Type up a Report in 12.5 Hours and Edit it in 7.5 Hours. GMAT Problem Solving
• How Many Litres of a 90% Solution of Concentrated Acid Needs to be Mixed with a 75% Solution GMAT Problem Solving GMAT Problem Solving
• What is the Largest Power of 3 Contained in 200! GMAT Problem Solving
• What is the Value of x? 1) x^2 + x + 10 = 16 2) x = 4y^4+2y^2+2 GMAT Problem Solving
• A Jar Full of Whisky Contains 40% Alcohol GMAT Problem Solving
• X is Older Than Y, Z Is Younger Than W And V is Older Than Y. Is Z Younger Than X? GMAT Problem Solving
• A Flagstaff 17.5 m High Casts a Shadow of Length 40.25 GMAT Problem Solving
• Frances Can Complete a Job in 12 hours, and Joan Can Complete the Same Job in 8 Hours GMAT Problem Solving
• Running at the Same Constant Rate, 6 Identical Machines can Produce a Total of 270 Bottles Per Minute. GMAT Problem Solving
• If abc ≠ 0 is (a/b)/c = a/(b/c) GMAT Problem Solving
• If 0 < a < b < c, Which of the Following Statements Must be True? GMAT Problem Solving
• Find The Value Of X GMAT Problem Solving
• Two Consultants, Mary and Jim, Can Type up a Report in 12.5 Hours and Edit it in 7.5 Hours GMAT Problem Solving
• The Value of k if the Sum of Consecutive Odd Integers From 1 to k Equals 441 GMAT Problem Solving
• A Circle with a Radius R is Inscribed into a Square with a Side K. GMAT Problem Solving
• Machine A can do a Certain Job in 12 Days Working 2 Full Shifts GMAT Problem Solving
• The square of 5√252 =? GMAT Problem Solving