DIY Challenge 3: Observing the pattern
What students worked out14
Student 9
1)Yes, there is a pattern in the number of ways.
If a 3rd ball is added to the existing mix of 2 balls, the new ball can be placed in 3 different positions. This means 3×2 or 6 positions in total.
If a 4th ball is added to the existing mix of 3 balls, the new ball can be placed in 4 different positions. This means 4x3x2 or 24 positions in total.
If a 5th ball is added to the existing mix of 4 balls, the new ball can be placed in 5 different positions. This means 5x4x3x2 or 120 positions in total.
And so on…. Please see the attached picture.2)
5 -> 120
6 -> 720Student 9
Attaching the image again
Student 4
If we have 4 balls and we take one of the balls and place it at the start, we will have 3 balls left. We have 6 possibilities for three balls. So we have six possibilities for each of the four balls if placed at the start. So the number of possibilities for the previous number gets multiplied by the number of balls which is one more than the previous number. That is why the pattern is occurring.
Karanteacher
Makes sense. Very clear and simple explanation.
Student 2
PATTERN 1
PUT ANY BALL FIRST eg: GREEN(g)
we can arrange the rest 3 in 6 ways
so we get six ways with each ball first
6×4
=24
THIS IS FOR 4 BALLS WE CAN USE IT FOR ANY
PATTERN 2
WE CAN GET THE NEXT NUMBER WITHOUT DOING IT COMPLETELY
WITH 1 BALL 1(A)
WITH 2 BALLS Ax2=2(B)
WITH 3 BALLS Bx3=6(C)
WITH 4 BALLS Cx4=24(D)
WITH 5 BALLS Dx5=120(E)
WITH 6 BALLS Ex6=720(F)
WITH 7 BALLS Fx7=5040(G)
WITH 8 BALLS Gx8=40320(H)
WITH 9 BALLS Hx9=362880(I)
WITH 10 BALLS Ix10=3628800(J)
shocking for ten balls for 15 balls it is
1307674368000Karanteacher
Wow….It’s interesting that how quickly the number of ways increase as we increase the number of balls.
Student 11
The pattern is like
We have to find the product of the no. of ways to arrange 4 and 5, which is 120.
Like that for six we have to find the product of the no. of ways to arrange 5 ang 6 which is 720.Karanteacher
Good..Very clearly explained. I think you also shared the reason in the previous challenge.
Student 18
1×2=2
2×3=6
6×4=24
therefore 24×5=120
and 120×6= 720Student 15
for 5=120
and for 6=720Student 14
1) The number of ways of the last number of balls
get’s multiplied with the next number of balls2) 24×5=120
120×6=720Student 3
120
7201×2
2×3
6×4
24×5
120×6Student 1
the pattern is whatever number times all the numbers before it like 3x2x1=6 and 4x3x2x1x=24 like that you can figure it out for any number.
for number 5 its 120
for number 6 its 720Student 17
6 taken 4=24 so 24 taken 5=120 and 120 taken 6=720