Recent questions tagged counting

nC2 for n=0,1,2,3...

How many ways $4$ boys and $4$ girls can sit in a row so that no two girl are sitting side by side

Given m integers $a_1,a_2,....,a_m$ show that there exist integers $k,s$ with $0 \leq k < s \leq m$ such that$a_{k+1} + a_{k+2}​​​​​​ + .....+a_s$ is divisible by $m$.

Suppose a graph $G$ has $6$ nodes. Prove that either $G$ or $G'$ must contain a triangle.($G'$ is the complement of $G$.) Prove it using pigeonhole principle.

Find the number of seven digit integers with sum of the digits equal to $11$ and formed by using the digits $1,2$ and $3$ only.Soln- $X_{1}+X_{2}+.......X_{7}=11$$(x+x^{2...

Show that, in a grid, the number of paths from $(0,0)$ to $(n,n)$ which does not cross ( it could touch ) the line $x = y$ is \begin{align*} \frac{1}{1+n}\binom{2\cdot n...

Using the letters PQRST , without repetition , how many 4 letters strings can be formed that,1) start or end with P..2) start with either P or Q.3) start with Q or end wi...

How many strings with seven or more characters can be formed from the letters of the word $\text{EVERGREEN}$ ?

One Hundred tickets, numbered $1,2,3,...,100$, are sold $100$ different people for a drawing. Four different prizes are awarded, including a grand prize(a trip to Tahiti)...

How many ways are there to put four different employees into three indistinguishable offices when each office can contain any number of employees?

How many bit strings of length $n$ contain exactly $r$ $1's$?

What is the number of ways to distribute ten identical sweets to three children?If each children gets at least one sweet.If 1 condition is omitted.P.S. I do not know the ...

Question How many different strings of length 2 can be made from the letters in 'ORONO'?I am getting 7 strings as follows: OO, ON, OR, NO, RO, RN, NRbut given answer is 6...