Divide into groups

Of 3n+1 objects, n are indistinguishable, and the remaining ones are distinct. Find the number of ways to choose n objects from them.

2 Answers

2305

Shaswata Roy ·2013-02-25 04:09:21

Is it-³ⁿ⁺¹C_n+³ⁿ⁺¹C_n-1+³ⁿ⁺¹C_n-2+.....+³ⁿ⁺¹C₀

2305

Shaswata Roy ·2013-02-25 04:23:36

If No. of indistinguishable objects = 0,#ways to select n objects = ²ⁿ⁺¹C_n

If No. of indistinguishable objects = 1,#ways to select rest of the objects = ²ⁿ⁺¹C_n-1

If No. of indistinguishable objects = 2,#ways to select rest of the objects = ²ⁿ⁺¹C_n-2

Hence the # ways is = ²ⁿ⁺¹C_n+ ²ⁿ⁺¹C_n-1+ ²ⁿ⁺¹C_n-2.....²ⁿ⁺¹C₀

Since this no. is equal to ²ⁿ⁺¹C_n+ ²ⁿ⁺¹C_n+1+ ²ⁿ⁺¹C_n+2.....²ⁿ⁺¹C_2n+1

Therefore answer is = ²ⁿ⁺¹C₀+ ²ⁿ⁺¹C₁+ ²ⁿ⁺¹C₂.....²ⁿ⁺¹C_2n+12 = 2ⁿ⁺¹/2 = 2ⁿ

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