76 lines
5.2 KiB
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76 lines
5.2 KiB
HTML
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<h2>Exploring Pascal's pyramid.</h2><div class="info" style="cursor:help;width:200px;margin-bottom:10px;"><h3>Problem 154</h3><span style="width:300px;color:#666;">Published on Saturday, 12th May 2007, 06:00 am; Solved by 1285</span></div>
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<p>A triangular pyramid is constructed using spherical balls so that each ball rests on exactly three balls of the next lower level.</p>
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<div style="text-align:center;"><img src="project/images/p_154_pyramid.gif" width="488" height="379" alt="" /></div>
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<p>Then, we calculate the number of paths leading from the apex to each position:</p>
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<p>A path starts at the apex and progresses downwards to any of the three spheres directly below the current position.</p>
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<p>Consequently, the number of paths to reach a certain position is the sum of the numbers immediately above it (depending on the position, there are up to three numbers above it).</p>
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<p>The result is <i>Pascal's pyramid</i> and the numbers at each level <var>n</var> are the coefficients of the trinomial expansion
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(<var>x + y + z</var>)<sup><var>n</var></sup>.</p>
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<p>How many coefficients in the expansion of (<var>x + y + z</var>)<sup>200000</sup> are multiples of 10<sup>12</sup>?</p>
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