The teacher wants to choose two of them to hand out books; in how many ways
can she choose a pair (ladja, 4)? 1.A & B 2.A & C 3.A & D 4.B & C 5.B & D 6.C &
D There are six ways to make a choice of a pair. If the teacher wants to send
three students: 1.A, B, C 2.A, B, D 3.A, C, D 4.B, C, D If the teacher wants to
send a group of K children where K may range from 0-4; in how many ways will she
choose the children K=0 1 way (There is only one way to send no children) K=1 4
ways ( A; B; C; D) K=2 6 ways (like above with Annie, Bob, Carlos, Danny) K=3 4
ways (above with triplets) K=4 1 way (there is only one way to send a group of
four) The above numbers (1 4 6 4 1) are the fourth row of numbers in Pascal
Triangle (Ladja, 5).
If we extend Pascal's triangle to infinitely many rows, and
reduce the scale of our picture in half each time that we double the number of
rows, then the resulting design is called self-similar -- that is, our picture
can be reproduced by taking an subtriangle and magnifying it, Granville
notes.The pattern becomes more evident if the numbers are put in cells and the
cells colored according to whether the number is 1 or 0 (Peterson's, 5).Similar,
though more complicated designs appear if one replaces each number of the
triangle with the remainder after dividing that number by 3. So, I get: 1 1 1 1
2 1 1 0 0 1 1 1 0 1 1 1 2 1 1 2 1 1 0 0 2 0 0 1 This time, one would need three
different colors to reveal the patterns of triangles embedded in the array. One
can also try other prime numbers as the divisor (or modulus), again writing down
only the remainders in each position (Freedman, 5). Actually, there's a simpler
way to try this out. With the help of Jonathan Borwein of Simon Fraser
University in Burnaby, British Columbia, and his colleagues, Granville has
created a Pascal's Triangle Interface on the web. One can specify the number of
rows (up to 100), the modulus (from 2 to 16), and the image size to get a
colorful rendering of the requested form.It's a neat way to explore the fractal
side of Pascal's triangle. Here's one example that I tried out, using 5 as the
modulus (Petetson's, 5).
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