IIT JEE , AIEEE Forum - Mathematics , Algebra

vineetnegi

I have a doubt

in order to find the no. of selection of 4 alphabets from the word INFINITE

I.................3

N...............2

F...............1

T...............1

E...............1

I tried multinomial theorem

the ans will be the coefficient of x^4

(1+x+x^2+x^3)(1+x+x^2)(1+x)(1+x)(1+x)

(1+x)^3(1 - x^4)[(1-x)^ -1](1 - x^3)[(1-x)^-1]......

(1+3x+3x^2+x^3)(1-x^3-x^4+x^7)(1+2x+3x^2+4x^3+5x^4.............

coefficients

x^4 taken.........5,-1

x^3,x taken..........-3,-2,2

x^2,x^2 taken..........9

if i add simply

5-1-3-2+2+9=10

but if i add by making them +

5+1+3+2+2+9=22

where as 22 is the ans in book

will 22 be the coefficient

please help ..........

so please tell what your ans is coming

akhil_o

my answer is coming 22..

coefficient of x⁴ in (1+x+x²+x³)(1+x+x²)(1+x)³
                              A                   B       C
now the possible distributions for selection are

A          B          C            coefficient

3           1          0                1
3           0          1                3
2           2          0                1
2           0          2                 3
0           1          3                1
1           0          3                1
0           2          2                3
2           1          1                3
1           2          1                3
1           1           2               3

see that whenever a selection is made from A or b, coefficient of the power of x is always 1
when we choose from C,
if r are chosen, coeff= 3Cr
so product is always equal to 3Cr, where r is the number chosen from C

so total selections=22

computer001

even i got 22..
but i din use all dis:
4 cases 0I 1 I 2I 3I
if there are 3i: then no of ways is 4
if 2i then num of ways is 1+3+3+1
if 1 i then num of ways is 3+3+1
if 0 i: 3

vineetnegi

I agree

thanks for resp.

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