# Counting problem

• Nov 17th 2008, 10:29 AM
bhuvan
Counting problem
Let S = {1,2,3,4,5}
(a) List all the 3-permutations of S
(a) List all the 3-combinations of S

Can any one help me with this ??
• Nov 17th 2008, 12:12 PM
TitaniumX
Quote:

Originally Posted by bhuvan
Let S = {1,2,3,4,5}
(a) List all the 3-permutations of S
(a) List all the 3-combinations of S

Can any one help me with this ??

a) 5C3 = 5!/(3!*(5-3)!) = 10 (order doesn't matter)
b) 5P3 = 5!/(5-3)! = 60 (order matters)

a)
{1,2,3}
{1,2,4}
{1,2,5}
{1,3,4}
{1,3,5}
{1,4,5}
{2,3,4}
{2,3,5}
{2,4,5}
{3,4,5}
Total = 10

b)
{1,2,3}
124
125
132
134
135
142
143
145
152
153
154
.
.
.
• Nov 17th 2008, 01:33 PM
Soroban
Hello, bhuvan!

Quote:

Let $S \:=\: \{1,2,3,4,5\}$

(a) List all the 3-permutations of $S$

There are $_5P_3 \:=\:\tfrac{5!}{2!} \:=\:60$ of them.

I'll start the list . . .

. . $\begin{array}{cccccccccccc}
123 & 124 & 125 & 132 & 134 & 135 & 142 & 143 & 145 & 152 & 153 & 154 \\ \\[-4mm]
213 & 214 & 215 & 231 & 234 & 235 & 241 & 243 & 245 & 251 & 253 & 254 \\ \\[-4mm]
312 & 314 & 315 & 321 & 324 & 325 & 341 & 342 & 345 & 351 & 352 & 354 \\
& & & & \hdots & \text{etc.} & \hdots\end{array}$

Quote:

(b) List all the 3-combinations of $S$
There are $_5C_3 \:=\:\tfrac{5!}{3!2!} \:=\:10$ of them . . .

. . $\begin{array}{ccccc}(1,2,3)&(1,2,4)&(1,2,5)&(1,3,4 )&(1,3,5) \end{array}$ . $\begin{array}{ccccc}(1,4,5)&(2,3,4)&(2,3,5)&(2,4,5 )&(3,4,5) \end{array}$

Edit: . Ha! TitaniumX beat me to it . . .
.
• Nov 17th 2008, 02:58 PM
bhuvan
thank you very much ...

How many solutions are there to the equation

x1+x2+x3+x4=17

where x1,x2,x3 and x4 are nonnegative integers ??

The number of ways to put N identical ones into k different variables (non-negative integers) is ${{N+k-1} \choose {N}}$.