sets

Question

Answer 1

Ans- D. 

No of elements in set = 2n + 1.

The number of subsets of the set which contains at most n elements = 

    ²ⁿ⁺¹C₀ + ²ⁿ⁺¹C1 + ²ⁿ⁺¹C2 + ²ⁿ⁺¹C3 + ................................+ ²ⁿ⁺¹Cn

= ( 2^{(2n + 1)}) / 2

= 2²ⁿ 

Note -ⁿ C₀+ ⁿC₁ + ............ⁿ C_n = 2ⁿ.

         ²ⁿ⁺¹C₀ + ²ⁿ⁺¹C1 + ²ⁿ⁺¹C2 + ²ⁿ⁺¹C3 + ................................+ ²ⁿ⁺¹Cn = 

²ⁿ⁺¹Cn+1 + ²ⁿ⁺¹Cn+2 + ²ⁿ⁺¹Cn+3 + ²ⁿ⁺¹Cn+4 + ................................+ ²ⁿ⁺¹C2n+1 =  ( 2^{(2n + 1)}) / 2

Comments

explain the second last line in the answer please.

---

⁵C₀ + ⁵C₁+ ⁵C₂ + ⁵C₃ + ⁵C₄ + ⁵C₅ = 2⁵

⁵C₀ + ⁵C₁+ ⁵C₂  =  ⁵C₃ + ⁵C₄ + ⁵C₅ = (2⁵)/2 = 2⁴.

---

Okay so you used nCr = nC(n-r). Thanks.

---

yes