Drawing letters

A string of alphabets are randomly generated one letter at a time. Each time, one obtains the letter $A,\cdots,Z$ with probability $p(A),\cdots, p(Z)$. Given that the sum of these probabilities is 1, what is the expected number of draws before the string "DHARMATH" appears?

Paired cards

A standard deck of 52 cards is shuffled with uniform probability. A "pair" is defined as two numerically identical cards that are adjacent in the stack. What is the probability that the deck contains no pair?

Challenge: Find the probability that the deck contains exactly $n$ pair for $n=1,2,\cdots,26$.

Infinite fractional sum

Find the limit to the infinite sum:

$$ 1 - \frac{1}{2} +\frac{1}{3} - \frac{1}{4} + \cdots$$

Solution: Coupon drawing

Original problem here: http://dharmath.thehendrata.com/2010/01/12/coupon-drawing/

A container holds $N$ coupons. You draw successive coupons from the container, observe the coupon drawn, and then replace the coupon. What is the expected number of draws needed until all $N$ coupons have been seen at least once?

Challenge: what is the probability that the process ends after exactly $M$ draws?

Coupon drawing

Credit for this problem goes to Sander Parawira

A container holds $N$ coupons. You draw successive coupons from the container, observe the coupon drawn, and then replace the coupon. What is the expected number of draws needed until all $N$ coupons have been seen at least once?

Solution here: http://dharmath.thehendrata.com/2010/01/12/solution-coupon-drawing/

Factorial and power of 2

Credit for this problem goes to Peter Macko.

If $n$ is a positive integer, prove that $(2^n)! $ is divisible by $2^{2^n-1}$ and that its quotient is an odd number.

Inequality

For positive numbers $a,b,c>0$ such that $a+b+c=3$, find the maximum value of

$\frac{1}{2+a^2+b^2} + \frac{1}{2+a^2+c^2} + \frac{1}{2+b^2+c^2} $