Direct sum of permutations
Imagine you have some toys, like blocks or cars or dolls. Now imagine that you want to organize them in a special way, but you don't want to mix them up. This is called a "permutation".
For example, let's say you have three toy cars - a red one, a blue one, and a green one. You can make different permutations of these cars by lining them up in different orders.
The first permutation could be:
1. Red car
2. Blue car
3. Green car
The second permutation could be:
1. Red car
2. Green car
3. Blue car
And so on.
Now, let's say you have some more toys - like some blocks and some dolls. You can make permutations of these toys, too.
The "direct sum" of permutations is when you take two different sets of toys and make permutations of them separately, but then put the two sets of permutations together.
For example, let's say you have three blocks - a yellow one, a purple one, and a red one - and two dolls - one with blonde hair and one with brown hair.
You can make permutations of the blocks:
1. Yellow block
2. Purple block
3. Red block
And permutations of the dolls:
1. Blonde doll
2. Brown doll
Then, you can put those two sets of permutations together in a new way:
1. Yellow block, blonde doll
2. Purple block, blonde doll
3. Red block, blonde doll
4. Yellow block, brown doll
5. Purple block, brown doll
6. Red block, brown doll
This is the direct sum of permutations! It's like you made two different towers of toys, and then stuck them together to make a big tower with both sets of toys.
For example, let's say you have three toy cars - a red one, a blue one, and a green one. You can make different permutations of these cars by lining them up in different orders.
The first permutation could be:
1. Red car
2. Blue car
3. Green car
The second permutation could be:
1. Red car
2. Green car
3. Blue car
And so on.
Now, let's say you have some more toys - like some blocks and some dolls. You can make permutations of these toys, too.
The "direct sum" of permutations is when you take two different sets of toys and make permutations of them separately, but then put the two sets of permutations together.
For example, let's say you have three blocks - a yellow one, a purple one, and a red one - and two dolls - one with blonde hair and one with brown hair.
You can make permutations of the blocks:
1. Yellow block
2. Purple block
3. Red block
And permutations of the dolls:
1. Blonde doll
2. Brown doll
Then, you can put those two sets of permutations together in a new way:
1. Yellow block, blonde doll
2. Purple block, blonde doll
3. Red block, blonde doll
4. Yellow block, brown doll
5. Purple block, brown doll
6. Red block, brown doll
This is the direct sum of permutations! It's like you made two different towers of toys, and then stuck them together to make a big tower with both sets of toys.