Question

Show that
C₁ + C₂ + C₃ + ... + C₆ = 63

Answer 1

Answer:

⁶C₁  + ⁶C₂  + ⁶C₃  + ⁶C₄  + ⁶C₅  + ⁶C₆   = 63

Step-by-step explanation:

⁶C₁  + ⁶C₂  + ⁶C₃  + ⁶C₄  + ⁶C₅  + ⁶C₆   = 63

ⁿCₓ  = n!/(x!(n-x)!)

LHS

= 6!/(1!(6-1)!)  + 6!/(2!(6-2)!) + 6!/(3!(6-3)!) + 6!/(4!(6-4)!) + 6!/(5!(6-5)!) + 6!/(6!(6-6)!)

= 6!/(1!5!) + 6!/(2!4!) + 6!/(3!4!) + 6!/(4!2!) + 6!/(5!1!) + 6!/(6!0!)

= 6 + 6*5/2  + 6*5*4/(3*2*1)  + 6*5/2  + 6  + 1

= 6 + 15 + 20 + 15 + 6 + 1

=63

= RHS

Hence

⁶C₁  + ⁶C₂  + ⁶C₃  + ⁶C₄  + ⁶C₅  + ⁶C₆   = 63

or we can say that

⁶C₁  + ⁶C₂  + ⁶C₃  + ⁶C₄  + ⁶C₅  + ⁶C₆ is

expansion of ( x + y) ⁶  where x & y = 1

= ( 1 + 1)⁶

= 2⁶

= 64