In a group of 6 boys and 4 girls, four children are to be selected. In how many different ways can they be selected such that at least one boy should be there?  Pick one: A) 209 B) 210 C) 211 D) 213 E) 215
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 options 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 × 4C1   We can select 2 boys and 2 girls ...(option 3)   Number of ways to this = 6C2 × 4C2   We can select 1 boy and 3 girls ...(option 4)   Number of ways to this = 6C1 × 4C3   Total number of ways   = 6C4 + 6C3 × 4C1 + 6C2 × 4C2 + 6C1 × 4C3   = 6C2 + 6C3 × 4C1 + 6C2 × 4C2 + 6C1 × 4C1[∵ nCr = nC(n-r)]   =   6   ×   5   2   ×   1   +   6   ×   5   ×   4   3   ×   2   ×   1   ×   4   =6×52×1+6×5×43×2×1×4   +   6   ×   5   2   ×   1   ×   4   ×   3   2   ×   1   +   6   ×   4   +6×52×1×4×32×1+6×4   =   15   +   80   +   90   +   24   =   209   209nswer is 209
<result>A</result>
In a group of 6 boys and 4 girls, four children are to be selected. In how many different ways can they be selected such that at least one boy should be there?