3 dolls, 4 kitchen sets and 3 cars are displayed in a toy shop. In how

1.	3 dolls, 4 kitchen sets and 3 cars are displayed in a toy shop. In how many ways can 3 toys be selected such that,(i) all are dolls?(ii) all are different toys?(iii) two are dolls and one is a kitchen set?
Answer» 3 dolls, 4 kitchen sets and 3 cars are displayed in a toy shop. 3 toys are to be selected. (i) All are dolls : There are 3 dolls. ∴ Total combinations = ³C₃ = 1 (ii) All are different toys: One doll out of 3 dolls can be selected in ³C₁ ways and one kitchen set out of 4 sets can be selected in 4CX ways and one car out of 3 cars can be selected in ³C₁ ways. ∴ Total combinations = ³C₁ × ⁴C₁ × ³C₁ =3x4x3 = 36 (iii) Two are dolls and one is a kitchen set: Two dolls out of 3 dolls can be selected in ³C₂ ways and one kitchen set out of 4 sets can be selected in ⁴C₁ ways. ∴ Total combinations = ³C₂ × ⁴C₁ = \(\frac{3×2}{2×1} × 4\) = 3 × 4 = 12

3 dolls, 4 kitchen sets and 3 cars are displayed in a toy shop. In how many ways can 3 toys be selected such that,(i) all are dolls?(ii) all are different toys?(iii) two are dolls and one is a kitchen set?

Answer»

3 dolls, 4 kitchen sets and 3 cars are displayed in a toy shop. 3 toys are to be selected.

(i) All are dolls : There are 3 dolls.
∴ Total combinations = ³C₃ = 1

(ii) All are different toys: One doll out of 3 dolls can be selected in ³C₁ ways and one kitchen set out of 4 sets can be selected in 4CX ways and one car out of 3 cars can be selected in ³C₁ ways.
∴ Total combinations = ³C₁ × ⁴C₁ × ³C₁ =3x4x3 = 36

(iii) Two are dolls and one is a kitchen
set: Two dolls out of 3 dolls can be selected in ³C₂ ways and one kitchen set out of 4 sets can be selected in ⁴C₁ ways.
∴ Total combinations = ³C₂ × ⁴C₁
= \(\frac{3×2}{2×1} × 4\)
= 3 × 4
= 12

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