If I am not getting your question wrong then your answers are wrong. a)If a 4 digit number contains exactly four 9's , then $^9C_4$ is not your solution , there is only one way i.e 9999 , so answer ...

Building up on your work :- You have found that $$2b>a+c... (1)$$$$2a>b+c...(2)$$Note that $ax^2+2bx+c$ has all positive coefficients , there fore the parabola of this quadratic equation will ...

I am not using the TeX codes , sorry . I hope you can understand this handwritten version.

There are obvious answers of the form $(n,1,1,1,)$ and its permutations as you have correctly pointed out.Let there be two numbers from $a,b,c,d$ greater than 1 , suppose a and b are greater than 1 ...

I thought a diagram will be useful here.As you can see I have drawn sets A & B and X & Y.The symmetric difference of A and B i.e. $A\Delta B$ is coloured in buff color(light orange).You have ...

Tchebychef's inequality-If $a_1,a_2..a_n$ and $b_1,b_2..b_n$ are real numbers then $$\frac{a_1b_1+a_2b_2+ .. +a_nb_n}{n}\geq\left(\frac{a_1+a_1+...+a_n}{n}\right)\left(\frac{b_1+b_1+...+b_n}{n}\right)... View answer 0 votes Obviously \sqrt{1+sin\alpha cos\alpha} will always be the longest side (always >1).Now we find \alpha for which this will be maximum.$$\frac{d}{d\alpha}\sqrt{1+sin\alpha cos\alpha}=\frac{cos(2\...

You can approach in the following way :-Take an elementary disc of infinitesimal thickness at a distance $x$ from the center of the face which is having the diameter.Calculating the linear density(I ...