Coaching Centre
Grouping, complementary and cases
Three more counting principles for ordered selections: treat objects that must be together as a block, count the complement, and split a problem into cases.
The grouping principle
Four boys and three girls stand in a line. In how many ways can this happen if the three girls stand together?
In how many ways can the four boys and three girls stand if boys and girls alternate?
The complementary principle
Seven people stand in a line. In how many ways can this happen if Ali and Bea are not next to each other?
The cases principle
How many whole numbers less than $1000$ can be formed from the digits $1, 2, 3, 4, 5$, with no repeated digit?
Four boys and three girls stand in a line with the three girls together. How many ways?
$5! \times 3!$.
How many ways if boys and girls alternate?
$4! \times 3!$.
Seven people line up. In how many ways are Ali and Bea not next to each other?
$7! - 2 \times 6!$.
How many numbers less than $1000$ can be formed from $1, 2, 3, 4, 5$ without repetition?
$5 + 20 + 60$.
For "at least one", the most efficient method is usually:
"None" is a single easy case.
Six people line up. In how many ways are two particular people next to each other?
$5! \times 2$.
Five people line up, with one particular person at either end. How many ways?
$2$ choices of end, then $4!$ for the rest.
When you split a counting problem into separate cases, you combine the counts by:
The cases do not overlap.
Four boys and four girls sit in a row, boys and girls alternating. How many ways?
Two patterns (BGBG... or GBGB...), each with $4! \times 4!$: $2 \times 576$.
Four boys and four girls sit in a row with all the boys together and all the girls together. How many ways?
$2$ orders for the blocks, times $4! \times 4!$.
New numbers every time. Keep going until you get five in a row.