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Maths-

General

Easy

Question

The coefficient of $x to the power of 3 end exponent y to the power of 4 end exponent z to the power of 5 end exponent$ in the expansion of $left parenthesis x y plus y z plus x z right parenthesis to the power of 6 end exponent$ is

70
60
50
None of these

The correct answer is: 60

We have,
$left parenthesis x y plus y z plus z x right parenthesis to the power of 6 end exponent equals not stretchy sum for r plus s plus t equals 6 of fraction numerator 6 factorial over denominator r factorial s factorial t factorial end fraction open parentheses x y close parentheses to the power of r end exponent open parentheses y z close parentheses to the power of s end exponent open parentheses z x close parentheses to the power of t end exponent$
$equals not stretchy sum for r plus s plus t equals 6 of fraction numerator 6 factorial over denominator r factorial s factorial t factorial end fraction x to the power of r plus t end exponent y to the power of r plus s end exponent z to the power of s plus t end exponent$
If the general term in the above expansion contains $x to the power of 3 end exponent y to the power of 4 end exponent z to the power of 5 end exponent comma$ then
$r plus t equals 3 comma blank r plus s equals 4$ and $s plus t equals 5$
Also, $r plus s plus t equals 6$
On solving these equations, we get
$r equals 1 comma blank s equals 3 comma blank t equals 2$
$therefore C o e f f i c i e n t blank o f blank x to the power of 3 end exponent y to the power of 4 end exponent z to the power of 5 end exponent equals fraction numerator 6 factorial over denominator 1 factorial 3 factorial 2 factorial end fraction equals 60$

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