What is sum 10C0-3(10C1)+9(10C2)-27(10C3)+...-59049(10C10)=?

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The sum 10C0-3(10C1)+9(10C2)-27(10C3)+... 59049(10C10) has to be determined.

10C0-3(10C1)+9(10C2)-27(10C3)+... 59049(10C10)

= 1 -3*10+9*45- 27*120 + 81*210 - 243*252 + 729*210 - 2187*120 + 6561*45 - 19683*10 + 59049*1

= 1 - 30 + 405 - 3240 + 17010 - 61236 + 153090 - 262440 + 295245 - 196830 + 59049

= 1024

**The required sum is 1024**

`^10C_0-^10C_1 3+ ^10C_2 3^2-...+^10C_10 3^10`

`^10C_0+^10C_1(-3)+^10C_2(-3)^2+...+^10C_10(-3)^10`

`=(1-3)^10=(-2)^10=2^10=1024`

**Sources:**

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