IIT JEE , AIEEE Forum - Mathematics , Algebra

elastiboysai

Heres d next one

This ones realy simple

A bag contains 21 balls of 6 different colours, there being 6 of the first colour, 5 of the second, 4 of the third and so on.

Find the number of different selections of of 6 balls that can be made from the bag

akhil_o

konichiwa2x

is it 259?

elastiboysai

Soln plz..

sidsgr88

is the ans 343????

konichiwa2x

There are balls of six different colours. (6+5+4+3+2+1 = 21).

Let

represent the number of balls chosen of different colours.

Then,

with the conditin

.

Using multinomial theorem, the number of different possible ways by which this selection can be made is equal to the co-efficient of

in

We can neglect the terms with powers greater than 6,

Again neglecting the terms with power greater than 6 we get,

Coefficient of

in the above expression is

elastiboysai

Ya Das Rt

Your soln seems 2 be d same as mine wich is

LET THE COLOURS BE Ci (0<i<=6)

BALLS OF COLOUR Ci : 6-i (0<i<6)

Let the no. of selections from Ci be aj

a1+a2+a3+a4+a5+a6=6

0<=aj<=6-j (0<j<=6)

Using multinomial theorom we have to find the coefficient of a^6
in (a^0+a^1+..a^p) (6<=p<=1)

Using formula for GP we get
(1-a^7)(1-a^6)(1-a^5)(1-a^4)(1-a^3)(1-a^2)(1-a)^-6

Neglecting higher powers of a(>6), and then simplifying we get
it as coeff of a^6 in
(1-a^2-a^3-a^4)(1-a)^-6

Hence coefficient of a^6 = 6+6-1C6 - 6+4-1C4 - 6+3-1C3 -

6+2-1C2
=259

I don think ders any oder method??

sboosy

well this method is there .

The sum can be split into cases

1)All of the same colour -1 way

2)5 of a colour , 1 of different colour - 2 * 5 = 10

3)4 of a colour , 2 of same different colour - 3*4 = 12

4)4 of a colour , 2 of different colours - 5C2 * 3 = 30

5)3 of a colour , 3 of another - 4C2 = 6

6)3 of a colour ,2 of another , 1 another = 4*4*4=64

7)3 of a colour , 3 of 3 different colours = 4*5C3 = 40

8)2 ,2 , 2 of 3 different colours = 5C3 = 10

9)2 of a colour ,4 of 4 different colours = 5*5C4 = 25

10)2 of a colour ,2 of another ,2 of 2 different other colours = 5C2 * 4C2 = 60

11)All 6 different = 1

so total = 259 ...

never attempt this in exam ..