Permutations And Combinations

Step 1: Recall the properties of an equivalence relation. An equivalence relation is reflexive, symmetric, and transitive.
Step 2: Since is transitive, we have:
Step 3: Because is also reflexive, every , which implies:
Step 4: Hence,
Step 5: Since itself is an equivalence relation, is also an equivalence relation.

Step 1: Simplify the expression inside the limit.
Step 2: For large values of , higher powers of become negligible.
Step 3: Rewrite the expression:
Step 4: Now use the standard limit: Here, after exact expansion.
Step 5: Therefore,

Step 1: If no three points are collinear, the total number of straight lines formed by joining 10 points is:
Step 2: Out of these 10 points, 6 points are collinear. The number of lines formed by these 6 collinear points taken two at a time is: But all these 15 pairs lie on one single straight line. Hence, instead of 15 distinct lines, only 1 line should be counted.
Step 3: Extra lines counted due to collinearity:
Step 4: Correct number of distinct straight lines: Final Answer (up to two decimal places):

Step 1: For positive integers, put . Then Number of ordered solutions for fixed sum is .
Step 2: Required range:
Step 3: Total: This is option (C).

Step 1: A 4 digit even number has the form: where .
Step 2: Digits satisfy:
Step 3: Maximum digit sum possible = . Therefore only combinations close to this extreme are possible.
Step 4: Enumerate partitions of 34 with last digit even constraint; checking feasible sets gives 7 numbers. Hence → (D).

Step 1: Understand the given conditions.

The number is a 5-digit telephone number.
It must start with the digits and .
No digit is repeated.
Step 2: Fix the starting digits. The first two digits are fixed as: So, we now need to fill the remaining 3 digits. Step 3: Count the remaining available digits. Digits available: to digits Digits already used: Step 4: Arrange the remaining digits. The remaining 3 positions can be filled using permutations of 8 digits taken 3 at a time: Step 5: Final conclusion. The total number of such telephone numbers is .