4 votes

How many 7 digit numbers have a digit sum of 11

You want to find the number of solutions of $x_1+x_2+x_3+x_4+x_5+x_6+x_7=11$ in non-negative integers with some constraints. The first constraint is that $x_1 \geq 1$ so instead we'll seek the number ...
Robert Shore's user avatar
  • 21.8k
2 votes

I do not understand why do we divide with the k! that is used in nCk when multiplying two combinations or more combinations that has the same k value

You need to be very clear in the concept for such problems. Basically it depends whether teams are labeled (eg team A, team B, etc) or unlabeled. A further very important point to note is that even ...
true blue anil's user avatar
1 vote

The interpretation of 'at least one' event in probability

The outcome in which the three balls drawn are, in descending order, $\{20, 19, 18\}$, is counted in the event $X = 20$, since this event explicitly includes the following outcomes: $$\color{red}{\{20,...
heropup's user avatar
  • 125k
1 vote

intuitively Understanding meaning of a Combinatorics problem to reach solution

For each natural $n$ let $f(n)$ be the number of steps to identify the correct switch among $n$ switches in the worst case. I claim that $f(n)=\lceil \log_2 n\rceil$, that is $f(n)$ is the smallest ...
Alex Ravsky's user avatar
  • 82.4k
1 vote

Determining the sign of the following permutation

Hint: Instead of trying to compute the disjoint cycles representation, remember $$ \operatorname{sign}(\sigma)=\prod_{i<j}\frac{\sigma(i)-\sigma(j)}{i-j}=(-1)^{\#\{(i,j)\mid i<j\text{ and }\...
user10354138's user avatar
  • 31.5k

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