If $x = (1111)_{2}$ , $y = (1001)_{2}$ and $z = (110)_{2}$ , then what is $x^{3} - y 3 - z 3 - 3 x yz$ equal to?

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This question was previously asked in

NDA-I (Mathematics) Official Paper (Held On: 13 Apr, 2025)

Answer 1

Calculation:

Convert x, y, and z from binary to decimal:

⇒ x = (1111)₂ = $1\cdot2^{3}+1\cdot2^{2}+1\cdot2^{1}+1\cdot2^{0}=15$

⇒ y = (1001)₂ = $1\cdot2^{3}+0\cdot2^{2}+0\cdot2^{1}+1\cdot2^{0}=9$

⇒ z = (110)₂ = $1\cdot2^{2}+1\cdot2^{1}+0\cdot2^{0}=6$

Now evaluate $x^{3}-y^{3}-z^{3}-3xyz $

$15^{3}-9^{3}-6^{3}-3\!\times\!15\!\times\!9\!\times\!6 = 3375-729-216-2430 = 0$

The result of the expression is 0. In binary, 0 is simply (0)₂.

Hence, the correct answer is Option 4.

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