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Permutations
The number of new words that can be formed by rearranging the letters of the word ‘ALIVE’ is ________?
The word 'ALIVE' has 5 distinct letters. The number of new words formed by rearranging these letters is the number of permutations of 5 distinct letters, which is 5! = 120.
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Speed And Time
In a kilometer race, A beats B by 50 meters or 10 seconds. What time does A take to complete the race?
Since A beats B by 50 meters or 10 seconds in a 1000 meter race, B takes 10 seconds more than A to finish. B covers 950 meters in the same time A covers 1000 meters. Thus, B's speed = 950 meters / (A's time + 10 seconds). A's speed = 1000 meters / A's time. Equating speeds: 950 / (t + 10) = 1000 / t. Solving gives t = 200 seconds for A.
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Algebra - Word Problems
A person ordered 5 pairs of black socks and some pairs of brown socks. The price of a black pair was thrice that of a brown pair. While preparing the bill, the bill clerk interchanged the number of black and brown pairs by mistake, which increased the bill by 100%. What was the number of pairs of brown socks in the original order?
Let the number of brown pairs be x. Price of one black pair = 3 times price of one brown pair. Original bill = 5 * 3p + x * p = 15p + xp. After interchange, bill = x * 3p + 5 * p = 3xp + 5p. Given the bill increased by 100%, so new bill = 2 * original bill. Thus, 3xp + 5p = 2(15p + xp) => 3x + 5 = 30 + 2x => x = 25. However, this contradicts the options. Rechecking the calculation: 3x + 5 = 2(15 + x) => 3x + 5 = 30 + 2x => x = 25. Since 25 is option D, the original answer was correct. The initial explanation was missing, so this clarifies the reasoning.
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Combinatorics
A group of 10 representatives is to be selected out of 12 seniors and 10 juniors. In how many different ways can the group be selected if it should have 5 seniors and 5 juniors?
The group must have exactly 5 seniors and 5 juniors. The number of ways to choose 5 seniors out of 12 is 12C5, and the number of ways to choose 5 juniors out of 10 is 10C5. The total number of ways is the product 12C5 * 10C5, which corresponds to option C.
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Ratio and Proportion
Sharjeel has a container with a mixture of wine and water in the ratio 4:1. Sharjeel spills some of the mixture by accident and then replaces the spilled amount with water of the same quantity. After this, the wine to water ratio becomes 3:2. How much water did Sharjeel add?
Initially, the ratio of wine to water is 4:1. After spilling some mixture and replacing it with water, the ratio changes to 3:2. By setting the total volume as 1 unit and using algebra to represent the spilled amount, solving the equation leads to the amount of water added being 1/4 of the total volume.
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Combinatorics
A student has to opt for 2 subjects out of 5 subjects for a course: Commerce, Economics, Statistics, Mathematics 1, and Mathematics 2. Mathematics 2 can be offered only if Mathematics 1 is also opted. The number of different combinations of two subjects which can be opted is ________?
Since Mathematics 2 can only be chosen if Mathematics 1 is also chosen, the pair (Mathematics 1, Mathematics 2) is valid. The other subjects can be chosen freely in pairs. The valid pairs are: (Commerce, Economics), (Commerce, Statistics), (Commerce, Mathematics 1), (Economics, Statistics), (Economics, Mathematics 1), (Statistics, Mathematics 1), and (Mathematics 1, Mathematics 2), totaling 7 combinations.
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Permutations
How many four-digit numbers can be formed using the digits {1, 3, 4, 5, 7, 9} without repetition of digits?
The number of four-digit numbers formed from 6 distinct digits without repetition is calculated by permutations: 6P4 = 6 × 5 × 4 × 3 = 360. Hence, option A is correct.
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Combinatorics
A two-member committee comprising one male and one female is to be constituted out of five males and three females. Among the females, Ms. A refuses to be a committee member if Mr. B is selected as the male member. How many different ways can the committee be constituted?
There are 5 males and 3 females. We need to form a committee of one male and one female. Ms. A refuses to be on the committee if Mr. B is selected as the male member. Total ways to select one male and one female without restriction = 5 × 3 = 15. Since Ms. A refuses to be on the committee if Mr. B is selected, the pair (Mr. B, Ms. A) is not allowed. Therefore, total allowed committees = 15 - 1 = 14. This matches option C.
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Mixture And Alligation
In what ratio must water be mixed with milk to gain 16 2/3% by selling the mixture at cost price?
A gain of 16 2/3% means the mixture is sold at 116 2/3% of the cost price of milk. Since water is free, the ratio of water to milk is 1:6 to achieve this gain when selling at the cost price of milk.
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Number Series
In a series of six consecutive even numbers, the sum of the second and sixth numbers is 24. What is the fourth number?
Let the first even number be x. Then the six consecutive even numbers are x, x+2, x+4, x+6, x+8, x+10. The sum of the second and sixth numbers is (x+2) + (x+10) = 2x + 12 = 24. Solving for x gives 2x = 12, so x = 6. The numbers are 6, 8, 10, 12, 14, 16. The fourth number is x+6 = 6 + 6 = 12. Therefore, the correct answer is B (12).