IIT JEE , AIEEE Forum - Mathematics , Algebra - Contest [swordfish #2]: Find ways to select people on circular table

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Sixteen people are arranged around a circular table. Find the no. of ways of selecting seven people so that no two of them are consecutive.

crazynishant

Total no. of ways of arranging 16 people around a round table is 15!

Therefore selecting 7 out of 15 would be ¹⁵C₇

_{Therfor answer is 15!/7!8!}

crazyhardik

16 people can be arranged on a circular table in (16-1)! i.e 15!

Now in selecting 7 people such that none is consecutive would be as following

First person can sit in either of the 16 chairs

Second person can sit in rest of the 13 chairs

Third in 11, Fourth in 9,Fifth in 7, Sixth in 5 and so on.

Therefore total arrangements of the following problem will be 16*13*11*9*7*5*3=2162160

magiclko

total no of ways of selecting 7 ppl frm 16 ppl is ¹⁶C₇ and no of ways of having 2 consecutive ppl is ¹⁶C₂ .... thrfore.. total no of ways is ¹⁶C₇- ¹⁶C₂= 11320

bhuvnesh2007

15,1,3,5,2,4,7,9,6,8,10,12,14,11,13,16

akanksha_2606

16 people can be arranged in 15! ways.

now there are 16 spaces for 7 people.

these 7 persons can be arranged in 16 spaces in¹⁶p₇ways.

therefore total no. of ways of arrangement=15!*¹⁶p₇

uday_zingtudor

¹⁶_^C2

uday_zingtudor

3*16=48

narayana

tibu

ans: total no of people=16.

reqd no=7 such that no two are consecutive

that leaves us with 8 people and so

no of selections=⁸C₇*15!

ravitej

first person can be selected in 16c1 ways.now two persons adjacent to him cannot be selected.out of the remaining 13 persons 6 are to be selected and 7 are not to be selected.Mark + marks for persons not to be selected and they(persons not tobe selected) can be partitioned in 8 ways.out of these 8 partitions 6 are to be selected,in 8c6 ways

(16c1* 8c6)/7==64

dividing by 7 implies the first person chosen may be any of the selected 7,i.e 7 times we get same selection.

ans==64

divya

15!-16/9

vishnu_srivastava

let the people sitting be in sequence + - + - + - + -.............+

so basically we have to select 7 signs (positive egative)out of respective number.

therefore answer should be ⁸c₇+⁸c₇

vishnu_srivastava

let us consider the case of selecting a sign(+ve/-ve)

out of + - + - ....+ where total signs are 17.

but nou 17^th sign(as it will become consicutive)

similarly we select people

therefore answer should be:

= 2 * ⁸c₇

santosh_0804

since 16 people

2 must be selected

but it is nothing but arranging 2 people in between 14 people round the table

hence 14 gaps

14p2=91

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