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Expand (1 + a)6 and verify by putting a = 2 on both sides.

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(1 + a)6

6C0 1 a0 + 6C1 . 1 . a1 + 6C2 . 1 . a2 + 6C3 . 1 . a3 + 6C4. 1 .a4 + 6C5 . 1 . a5 + 6C6 1 . a6

= 1 + 6a + 15a2 + 20a3 + 15a4 + 6a5 + a6

LHS = (1 + a)6

Putting a = 2.

LHS = (1 + 2)6 = (3)6 = 729

RHS = 1 + 6a + 15a2 + 20a3 + 15a4 + 6a5 + a6

Putting a = 2.

RHS= 1 + 6 × 2 + 15(2)2 + 20(2)3 + 15(2)4 + 6(2)5 + (2)6

= 1 + 12 +(15 × 4) + (20 × 8) + (15 × 16) + 6(32) + 64

= 13 + 60 + 160 + 240 + 192 + 64

= 729

Hence, LHS = RHS

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