A triangle is to be constructed in the xy-plane such that the x- and y- coordinates of each vertex are integers that satisfy the inequalities -3 ≤ x < 7 and 2 < y ≤ 7. If one of the sides is parallel to the x-axis, how many different triangles with these specifications can be constructed?
Given:
To find: Number of different triangles that may be specified.
Approach:
Working Out:
Looking at the answer choices, we see that the correct answer is Option B
The players for a Tennis Mixed Doubles match are to be chosen from among 5 men and 4 women. In how many ways can the teams for the match be formed?
Given:
To find: Number of ways to choose the 2 teams for a Tennis Mixed Doubles Match
Approach:
So, Number of ways in which the players can be selected
= (Ways to Select 2 men out of 5)*(Ways to select 2 women out of 4)*(Ways to arrange the 2 selected men and 2 selected women into 2 teams)
Working Out:
Looking at the answer choices, we see that the correct answer is Option D
A soccer team of 11 players is to be formed out of a pool of 23 available players. The team should consist of 3 strikers, 4 midfielders, 3 defenders and 1 goalkeeper. The pool of 23 players consists of 6 strikers, 7 midfielders, 7 defenders and 3 goalkeepers. If Bruce, who is a striker, and David, who is a midfielder, should be a part of the team, what is the number of possible ways in which the team can be formed?
Given
To Find: Number of ways of forming the team?
Approach
3. As it’s an AND event,
The total number of ways to select the team = Task-1 * Task-2 * Task-3 * Task-4
Working Out
Thus, the team can be formed in 21000 ways.
Answer: D
How many 3-letter words can be formed using the letters from the word GALE, if repetition is not allowed?
Given: 4-letter word GALE
To find: Number of 3-letter words that can be formed without repeating any letters
Approach:
We will use the method of filling spaces to answer the question
Working Out:
Answer: Option E
How many 4 digit numbers greater than 4000 can be formed using the digits from 0 to 8 such that the number is divisible by 4?
Given
To Find: 4 digit integers > 4000 and divisible by 4?
Approach
Working Out
Answer: D
A company interviewed 5 applicants each for the posts of the Director and the President. If Jack and Jill were the only applicants who were interviewed for both the posts and an applicant can be selected for only one of the posts, what is the number of ways in which the company can select its Director and President from the interviewed applicants?
Given
To Find: Number of ways to select the Director and the President?
Approach
Working Out
Hence, there are 23 ways in which the company can select its director and the president.
Answer: B
How many positive 4-digit integers are divisible by 20 if the repetition of digits is not allowed?
Given:
To find: Number of positive 4-digit numbers that are divisible by 20
Approach:
Working Out:
Looking at the answer choices, we see that the correct answer is Option C
A salesman has five chocolates each of three different varieties. If he has to sell 9 chocolates to 9 different people and he can sell at most two varieties of chocolates, in how many different ways can he sell the chocolates?
Given
To Find: Number of ways in which the chocolates can be sold?
Approach
Working Out
Thus, there are 756 ways in which the shopkeeper can distribute the chocolates.
Answer: B
The figure above shows the different alternate routes possible to go from point A to point E. The arrows show the allowed direction of motion on each route. What is the total number of ways to go from point A to E?
Given: A figure with multiple routes to go from A to E
To find: Total number of ways to go from A to E
Approach:
Working Out:
Answer: Option B
The editor of an anthology of short stories has to select authors from a pool of 12 authors, of which 4 are female. If the anthology is to include the work of 8 authors of which at least 2 must be female, in how many ways can the editor select the authors?
Given:
To find: Number of ways in which this selection may be made
Approach:
Working Out:
Looking at the answer choices, we see that the correct answer is Option D
The figure above shows a rectangle ABCD in the xy- coordinate plane. The sides AB and AD are parallel to the x- and the y- axis respectively. How many squares of side 1 unit that lie on or inside the rectangle ABCD can be drawn?
(1) The length of side AB is 6 units
(2) The coordinates of points A and C are (3,2) and (9, 5)
Steps 1 & 2: Understand Question and Draw Inferences
Given: Rectangle ABCD in which AB is parallel to the x-axis and AD is parallel to the y-axis
To find: Number of squares of side 1 unit that can be drawn inside or on the rectangle ABCD
Let’s find the Number of squares of side 1 unit that can be formed in the same row as square APQR (let’s call this number r):
Similarly, we can find that the Number of squares that can be formed in the same column as square APQR (let’s call this number c) =
if AD is an integer
(Total number of squares that can be formed in the rectangular region ABCD) = r*c
Since the values of r and c depend on AB and AD, we need to find the values of AB and AD, that is, the length and breadth of rectangle ABCD.
Step 3: Analyze Statement 1 independently
Statement 1 says that ‘The length of side AB is 6 units’
Step 4: Analyze Statement 2 independently
Statement 2 says that ‘The coordinates of points A and C are (3,2) and (9, 5)’
Step 5: Analyze Both Statements Together (if needed)
Since we’ve already arrived at a unique answer in Step 4, this step is not required
Answer: Option B
The lowest integer that has both positive integers x and y as its factors can be written in the form of If x can be written as
, where a, b, c and d are positive integers, what are the possible number of values that y can take ?
LCM(x, y) =
To Find: Number of values y can take?
Approach
, but it can take only one value of prime factor 3, which is 3^{4}
4. So, number of possible values of y = Number of ways in which 2 can be present in y * Number of ways in which 3 can be present in y * Number of ways in which 5 can be present in y* Number of ways in which 7 can be present in y
Working Out
and
2. Prime factors for which their powers in LCM (x, y) is greater than that in x
and 7 do not have the same powers in LCM (x, y) and x
3. Prime factors for which their powers in LCM (x, y) and x are same
a. 3 and 5 have the same powers in both LCM (x, y) and x
, i.e
a total of (2b+c+1) ways………(3)
2. Possible number of ways in which 5 can be present in y =
, i.e. a total of (c+1) ways…….(4)
4. So, number of possible values of y = 1 * 1 * (2b+c+1) * (c+1) = (2b+c+1) (c+1)
So, there can be (2b+c+1) (c+1) values of y.
A code is formed by combining one of the letters from A- Z and two distinct digits from 0 to 9 such that if the letter in the code is a vowel, the sum of the digits in the code should be even and if the letter in the code is a consonant, the sum of the digits in the code should be odd. If the code is case-sensitive (for example, A is considered to be different from a), how many different codes are possible?
Given
To Find: Number of possible codes?
Approach
4. Case-II: If the letter selected is a consonant
5. Total number of ways of forming a code = Number of codes formed in Case-I + Number of codes formed in Case-II
Working Out
Case-II: If the letter is a consonant
Number of ways in which 1 letter and 2 digits can be arranged = 3!
Answer: D
Micky has 10 different letters and 5 different envelopes with him. If he can only put one letter in an envelope, in how many ways can the letters be put in the envelopes?
Given
To Find: Number of ways in which 5 letters can be put in 5 envelopes?
Approach
Working Out
In how many ways can 12 different books be distributed equally among 4 different boxes?
Given:
To find: Number of ways to distribute 12 different books equally among 4 different boxes.
Approach:
Working Out:
Looking at the answer choices, we see that the correct answer is Option C
The host of a television debate show has to select a 4-member discussion panel out of a list of 22 willing candidates that includes 5 politicians and 6 businessmen. If the list includes candidates from at least 4 professions and no two members of the discussion panel are to be of the same profession, then in how many ways can the panel be constituted?
(1) The list includes 5 journalists and 2 authors
(2) The list includes only 1 profession from which there are fewer than 3 candidatesStatement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
Steps 1 & 2: Understand Question and Draw Inferences
Given:
To find: Number of ways to constitute a 4 – member panel, such that all members have different professions
(RequiredNumberofways)=
Step 3: Analyze Statement 1 independently
So, since we have not been able to determine a unique number of professions, Statement 1 alone is not sufficient.
Step 4: Analyze Statement 2 independently
We quickly see that multiple values for the number of professions and for the number of people in each profession are possible here.
So, Statement 2 is not sufficient to answer the question
Step 5: Analyze Both Statements Together (if needed)
Since we now know the number of professions and the number of candidates in each profession, we will be able to answer the Q.
The 2 statements together are sufficient to answer the question.
Answer: Option C
What is the value of even positive integer n?
(1) The number of ways to choose 2 items out of n distinct items is 28
(2) The number of ways to choose 2 prime numbers out of the first n/2 positive integers is 1
Steps 1 & 2: Understand Question and Draw Inferences
Given: Even Integer n > 0
To find: n = ?
Step 3: Analyze Statement 1 independently
Statement 1 says that ‘The number of ways to choose 2 items out of n distinct items is 28’
Step 4: Analyze Statement 2 independently
Statement 2 says that ‘The number of ways to choose 2 prime numbers out of the first n/2
positive integers is 1’
Step 5: Analyze Both Statements Together (if needed)
Since we’ve already arrived at a unique answer in Step 3, this step is not required
Answer: Option A
In how many ways can 10 software engineers and 10 civil engineers be seated in a round table so that they are positioned alternatively?
The 10 civil engineers can be arranged in a round table in
(10-1)! = 9! Ways ---(A)
Now we need to arrange software engineers the round table such that software engineers
and civil engineers are seated alternatively. i.e., we can arrange 10 software engineers
in the 10 positions marked as * as shown below
This can be done in 10P10 = 10! Ways ---(B)
From (A) and (B),
The required number of ways = 9! × 10!
How many numbers not exceeding 10000 can be made using the digits 2,4,5,6,8 if repetition of digits is allowed?
Given that the numbers should not exceed 10000
Hence numbers can be 1 digit numbers or 2 digit numbers or 3 digit numbers
or 4 digit numbers
Given that repetition of the digits is allowed.
A. Count of 1 digit numbers that can be formed using the 5 digits (2,4,5,6,8) (repetition allowed)
The unit digit can be filled by any of the 5 digits (2,4,5,6,8)
Hence the total count of 1 digit numbers that can be formed using the 5 digits (2,4,5,6,8) (repetition allowed) = 5 ---(A)
B. Count of 2 digit numbers that can be formed using the 5 digits (2,4,5,6,8) (repetition allowed)
Since repetition is allowed, any of the 5 digits(2,4,5,6,8) can be placed
in unit place and tens place.
Hence the total count of 2 digit numbers that can be formed using the 5 digits (2,4,5,6,8) (repetition allowed) = 52 ---(B)
C. Count of 3 digit numbers that can be formed using the 5 digits (2,4,5,6,8) (repetition allowed)
Since repetition is allowed, any of the 5 digits (2,4,5,6,8) can be placed in unit place , tens place and hundreds place.
Hence the total count of 3 digit numbers that can be formed using the 5 digits (2,4,5,6,8) (repetition allowed) = 53 ---(C)
D. Count of 4 digit numbers that can be formed using the 5 digits (2,4,5,6,8) (repetition allowed)
Since repetition is allowed, any of the 5 digits (2,4,5,6,8) can be placed
in unit place, tens place, hundreds place and thousands place
Hence the total count of 4 digit numbers that can be formed using the 5 digits (2,4,5,6,8) (repetition allowed) = 5^{4} ---(D)
From (A), (B), (C), and (D),
total count of numbers not exceeding 10000 that can be made using the digits 2,4,5,6,8 (with repetition of digits)
= 5 + 5^{2} + 5^{3} + 5^{4}
In how many ways can 7 identical balls be distributed in 5 different boxes if any box can contain any number of balls?
Here n = 5, k = 7. Hence, as per the above formula, the required number of ways
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