In a group of 6 boys and 4 girls, four children are to be selected such that
at least one boy should be there.
Hence we have 4 choices as given below
We can select 4 boys ------(Option 1).
Number of ways to this = ⁶C₄
We can select 3 boys and 1 girl ------(Option 2)
Number of ways to this = ⁶C₃ x ⁴C₁
We can select 2 boys and 2 girls ------(Option 3)
Number of ways to this = ⁶C₂ x ⁴C₂
We can select 1 boy and 3 girls ------(Option 4)
Number of ways to this = ⁶C₁ x ⁴C₃
Total number of ways
= (⁶C₄) + (⁶C₃ x ⁴C₁) + (⁶C₂ x ⁴C₂) + (⁶C₁ x ⁴C₃)
= (⁶C₂) + (⁶C₃ x ⁴C₁) + (⁶C₂ x ⁴C₂) + (⁶C₁ x ⁴C₁) [Applied the formula ⁿC_r = ⁿC_{(n - r)} ]

$=\left[\frac{6\times 5}{2\times 1}\right]+\left[\left(\frac{6\times 5\times 4}{3\times 2\times 1}\right)\times 4\right]+\left[\left(\frac{6\times 5}{2\times 1}\right)\left(\frac{4\times 3}{2\times 1}\right)\right]+\left[6\times 4\right]$

= 15 + 80 + 90 + 24
= 209