The number of positive integral solutions of the equation x - y - z - - - 15- where x - 1- y - 2- z - 3 and - 4 is

Let nbsp; x = x_1 + 1 ⇒ x_1 ≥ 1 z=z_1 + 3 ⇒ z_1 ≥ 1 y = y_1 + 2 ⇒ y_1 ≥ 1 x + y + z + ω = 15 ⇒ x_1 + y_1 + z_1 = 9 ω So no. of positive integral solution ...

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Standard XII

Mathematics

Multinomial Expansion

The number of...

Question

The number of positive integral solutions of the equation $x + y + z + ω = 15$ , where $x > 1, y > 2, z > 3$ and $ω < 4$ is

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Solution

$Let x = x_{1} + 1 \Rightarrow x_{1} \geq 1 z = z_{1} + 3 \Rightarrow z_{1} \geq 1$
$y = y_{1} + 2 \Rightarrow y_{1} \geq 1$
$x + y + z + ω = 15 \Rightarrow x_{1} + y_{1} + z_{1} = 9 - ω$
So no. of positive integral solution
$= \sum_{ω = 1}^{3}$ $^{8 - ω} C_{2}$
$=^{7} C_{2} +^{6} C_{2} +^{5} C_{2} = 21 + 15 + 10 = 46$

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The number of positive integral solutions of the equation x+y+z+ω=15, where x>1,y>2,z>3 and ω<4 is

Let x=x1+1⇒x1≥1z=z1+3⇒z1≥1 y=y1+2⇒y1≥1 x+y+z+ω=15⇒x1+y1+z1=9−ω So no. of positive integral solution =∑3ω=1 8−ωC2 =7C2+6C2+5C2=21+15+10=46

The number of positive integral solutions of the equation $x + y + z + ω = 15$ , where $x > 1, y > 2, z > 3$ and $ω < 4$ is