My Math Forum statistics Que's

 March 15th, 2010, 10:17 AM #1 Newbie   Joined: Mar 2010 Posts: 2 Thanks: 0 statistics Que's hey ppl, here r da 2 statistics que's....... Q1) Calculate the number of ways in which the letters of the word TRIANGLES can be arranged if no two vowels may come together. Q2) How many 7-digit numbers can be formed from the digits 0,1,2,2,3,3,3 assuming that a number can't start with 0 ?????????
March 15th, 2010, 01:21 PM   #2
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Re: statistics Que's

Hello, maria1!

Quote:
 1) Calculate the number of ways in which the letters of the word TRIANGLES [color=beige]. . [/color]can be arranged if no two vowels may come together.

$\text{There are 6 consonents: }\:\{G,\ L,\ N,\ R,\ S,\ T\}$
[color=beige]. . [/color]$\text{and 3 vowels: {\:\{A,\ E,\ I\}$

$\text{Place the 6 consonants in a row, leaving a space before, after and between them.}$

[color=beige]. . [/color]$-\;c\; -\;c\; -\;c\;-\;c\;-\;c\;-\;c\;-$

$\text{The consonants can be placed in }\,6!\,=\,720\text{ ways.}$

$\text{Place the 3 vowels in 3 of the 7 spaces.}$
[color=beige]. . [/color]$\text{There are: }\:_7P_3 \,=\,210\text{ ways.}$

$\text{Therefore, there are: }\:710\,\times\,210 \:=\:151,200\text{ ways.}$

Quote:
 2) How many 7-digit numbers can be formed from the digits 0,1,2,2,3,3,3 [color=beige]. . [/color]assuming that a number can't start with 0 ?

$\text{With no restrictions, there are: }\:\frac{7!}{2!\,3!} \,=\,420\text{ possible numbers.}$

$\text{How many of them begin with zero?}$

$\text{If the first digit is 0, the other 6 digits }\{1,\,2,\,2,\,3,\,3,\,3\}\text{ can be arranged in : }\:\frac{6!}{2!\,3!} \,=\,60\text{ ways.}$
[color=beige]. . [/color]$\text{Hence, there are 60 numbers that begin with 0.}$

$\text{Therefore, there are: }\:420\,-\,60\:=\:360\text{ numbers that do }not\text{ begin with 0.}$

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