If $3^{a} + 3^{b} = 756, 7^{a} + 2^{c} = 375, 5^{a} + 3 = 128$ , then the value of $a + b + c$ is

12

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14

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18

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20

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Solution

The correct option is D $14$
$A)$ Given
$3^{a} + 3^{b} = 756 - - - (1)$
$7^{a} + 2^{c} = 375 - - - (2)$
$5^{a} + 3 = 128 - - - (3)$

solving $(3)$
$5^{a} + 3 = 128$
$= 5^{a} = 125$
$= 5^{a} = 5^{3}$
$∴ a = 3$

Substituting $a = 3$ in $(2)$
$7^{a} + 2^{c} = 375$
$= 7^{2} + 2^{c} = 375$
$= 343 + 2^{c} = 375$
$= 2^{c} = 375 - 343$
$= 2^{c} = 32$
$2^{c} = 2^{5}$
$= c = 5$

Substituting $a = 3$ in $(1)$
$3^{a} + 3^{b} = 756$
$= 3^{3} + 3^{b} = 756$
$= 3^{b} = 756 - 27$
$= 3^{b} = 729$
$= b = 6$
$∴ a + b + c = 3 + 5 + 6 = 14$

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