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10 December 2006

Powers of 11 in Pascal's Triangle

So one of the first things you might recognize are the first few rows... The numbers "feel" familiar...

That could be because they are actually powers of 11...
110 = 1
111 = 11
112 = 121
113 = 1331
114 = 14641

But what is 115? Whoa - hold up. Let's recap just a little...

Let's re-examine 112....
What we are really seeing is:



















102101100
121
Which is really...
1 * 1022 * 1011 * 100
100201
or... 100+20+1 = 121 = 112


The same is true for 113...






















103102101100
1331
Which is really...
1 * 1033 * 1023 * 1011 * 100
1000300301
or... 1000+300+30+1 = 1331 = 113


Now, 115 is a bit more tricky... The math is the exact same, but it is not as visually obvious...




























105104103102101100
15101051
Which is really...
1 * 1055 * 10410 * 10310 * 1025 * 1011 * 100
10000050000100001000501
or... 100000+50000+10000+1000+50+1 = 161051 = 115
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