The number of solutions to x+y+z=10, where 1≤x,y,z≤6 and x,y,z∈N, is equal to

Correct answer: (C) 27. x+y+z=10 Number of solutions is equal to the coefficient of x^10 in the expansion of x+x^2+x^3+.……+x^63=x^31-x^61-x3 or the coefficient of x^7 in the expansion of 1-x^631-x^(-3)=1-3x^61-x^(-3) =C_(7+3-17)-3C_(1+3-11) =C_97-3C_31=36-9=27

Q42 / 98 · NTA Abhyas (JEE Main) 2020 · Mathematics · Permutation Combination

The number of solutions to $x + y + z = 10$ , where $1 \leq x, y, z \leq 6$ and $x, y, z \in N$ , is equal to

A
$35$
B
$36$
Correct answer
C
$27$
D
$66$

Solution

x + y + z = 10

Number of solutions is equal to the coefficient of

x^{10}

in the expansion of

{(x + x^{2} + x^{3} + . \dots \dots + x^{6})}^{3} = x^{3} {(\frac{1 - x^{6}}{1 - x})}^{3}

or the coefficient of

x^{7}

in the expansion of

{(1 - x^{6})}^{3} {(1 - x)}^{- 3} = (1 - 3 x^{6}) {(1 - x)}^{- 3}

= {{}^{7 + 3 - 1}C}_{7} - 3 ({{}^{1 + 3 - 1}C}_{1})

= {{}^{9}C}_{7} - 3 ({{}^{3}C}_{1}) = 36 - 9 = 27

Hence, the correct answer is option: 'C'.

The number of solutions to x+y+z=10, where 1≤x,y,z≤6 and x,y,z∈N, is equal to

Solution

More Permutation Combination Questions

The number of solutions to $x + y + z = 10$ , where $1 \leq x, y, z \leq 6$ and $x, y, z \in N$ , is equal to