Exactly two were born on the same day:

Let's find the numerator first. Numerator is the total number of favorable events.
We can select 2 persons from 7 persons in $$^7C_2$$ ways and 1 day from the seven days in 7 ways => $$^7C_2$$ * 7
There are 5 persons remaining and 6 days remaining. Each of them was born on a different day.
So, we can select 5 days from 6 days in $$^6C_5$$ ways and arrange the 5 persons in them in 5! ways => $$^6C_5$$*5!
Total number of favorable events = $$^7C_2$$ * 7 * $$^6C_5$$*5!

Total number of events = $$7^7$$

Probability required = $$\frac{^7C_2 * 7 * ^6C_5 * 5!}{7^7}$$

Hi Sahil, the best way is to draw the graph in case you are solving questions on areas.

Hi Pulkit, thanks for your comment. Let's continue the discussion on that same thread.

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Hi Udita, the best way to solve percentage questions is to reduce them to fractions and then form equations based on the conditions given. The equations thus formed are generally linear or quadratic and can be easily solved. Finally, convert the fractions into percentages to get the answer.

They are out.

There is no one best way to solve Seating Arrangement problems but one good way to solve is to try to start from the piece of information that gives rise to the least number of possibilities. From this, include the information provided by all the other clues. One has to be careful about all the possibilities because many Seating Arrangement questions include the option 'Cannot be determined'.

8 and 131 are co-prime and Euler's Totient of 131 is 130.
=> $$8^{130}$$ mod 131 = 1
=> (8 mod 131) * ($$8^{129}$$ mod 131) = 1 mod 131
=> 8 * ($$8^{129}$$ mod 131) = 131k + 1
If p is equal to $$8^{129}$$ mod 131, then
=> 8p = 131k + 1
This has a solution at k = 5. At k = 5, the value of p is 82.
We know that p is equal to $$8^{129}$$ mod 131.
Hence, $$8^{129}$$ mod 131 = 82

By Wilson's theorem, 72! mod 73 = 72
72 mod 73 = -1
71 mod 73 = -2 and so on
So, 72! mod 73 = $$36!^2$$ mod 73 = 72

=> $$t^2$$ mod 73 = 72
=> $$t^2$$ + 1 = 73k
This satisfied for t = 27
So, 27 is the remainder of 36! mod 73

This is the solution given by Cracku - is this correct? I don't understand how 1+30r is getting changed to 1+10r

Let the rate of fall be rSo, 45(1+r) 30 = 20 => (1+r) 30 =0.44 Approximate the expression as 1+30r => 1+10r = 0.44 => r = -0.56/10 = -0.056 So, the rate of fall is 5.6%