## Thursday, January 21, 2010

### 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?

Labels:
Combinatorics,
expected value,
probability,
Random,
sequence,
string

## Wednesday, January 13, 2010

### 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 $n=1,2,\cdots,26$.

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

Labels:
card,
Combinatorics,
pair,
probability,
Random,
Solved

### Infinite fractional sum

Find the limit to the infinite sum:

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

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

Labels:
Algebra,
calculus,
infinite series,
infinite sum,
limit

## Tuesday, January 12, 2010

### 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?

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?

Labels:
Combinatorics,
drawing,
expected value,
Random,
repetition,
reset,
Solution

## Monday, January 11, 2010

### 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/

Labels:
Combinatorics,
drawing,
expected value,
Random,
replacement,
Solved

## Saturday, January 9, 2010

### 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.

Labels:
divisibility,
factorial,
Number Theory,
power of two

### 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} $

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

Labels:
3 variable,
constraint,
Inequality,
maximum,
symmetric

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