Permutations

To solve this problem, we use the concept of alligation. Alligation is a method used to find the ratio in which two or more ingredients at different prices are mixed to produce a mixture of a desired price.

Given:
Price of peas (C1) = ₹16/kg
Price of soyabean (C2) = ₹25/kg
Price of mixture (Cm) = ₹19/kg

Using the rule of alligation, we calculate the differences between the prices:

Difference between C2 and Cm:
(C2 - Cm) = 25 - 19 = 6

Difference between Cm and C1:
(Cm - C1) = 19 - 16 = 3

The ratio of peas to soyabean is therefore given by the differences:

Ratio = (C2 - Cm) : (Cm - C1) = 6 : 3

Simplifying this ratio gives:

6:3 = 2:1

Therefore, the shopkeeper must mix the peas and soyabean in the ratio of 2:1.

To determine how many 3-letter words can be formed from the letters of the word 'OBJECTS' without repetition, we start by noting the total number of unique letters. 'OBJECTS' consists of 7 distinct letters: O, B, J, E, C, T, S.

The formula for permutations, where no repetition is allowed, is given by:

where n is the total number of items to choose from, and r is the number of items to choose.

In this problem, n = 7 (the total number of unique letters) and r = 3 (since we are forming 3-letter words).

Substituting these values into the formula, we have:

P(7, 3) = 7! / (7-3)! = 7! / 4!

Calculate 7! and 4!:

Now, divide 5040 by 24:

7! / 4! = 5040 / 24 = 210

Thus, the total number of 3-letter words that can be formed is 210.

In conclusion, the correct answer is 210, not 840 as mentioned earlier in the options.

To determine the number of injective mappings from set A to set B, we should understand what an injective function is. An injective function, or one-to-one function, maps distinct elements of set A to distinct elements of set B.

Let's suppose set A has elements and set B has elements. For an injective function to exist, it must be that .

Given: Set A has elements and Set B has 6 elements. To find the number of injective mappings, we use the formula for permutations because each element from A has to map to a uniquely distinct element in B.

The number of injective mappings (or permutations) of choosing items from items is given by the permutation formula:

In this case, assume Set A also has elements:

Since the number of injective mappings equals 360:

The factorial of 6 is 720, so we simplify:

Solving for :

By evaluating factorial values:

, which matches our result, implying .

Solve for :

Thus, set A should have 4 elements for there to be 360 injective mappings from set A to set B.

Therefore, the number of injective mappings that can be defined from Set A to Set B is:

360

To solve the problem of selecting a committee consisting of 4 men and 3 women out of 6 men and 5 women, we use combinations since the order of selection does not matter. The combination formula is given by:

First, calculate the number of ways to select 4 men from 6 men:

Next, calculate the number of ways to select 3 women from 5 women:

The total number of ways to form the committee is the product of the two combinations:

Hence, there are 150 ways to select 4 men and 3 women to form the committee.

To determine how many 3-digit even numbers can be formed from the digits 0-9 with repetition allowed, follow these steps:

Therefore, there are 450 possible 3-digit even numbers formed from the digits 0-9 with repetition allowed.