﻿ Mathcounts Answer Keys

# Sprint 17466 and solutions

April 9, 2019

### Monday April 8 sprint.

##### Problem 1:
There are 10! ways to arrange the 10 items. Since 9 items aren't distinct, divide the 10! ways by the 9! ways of arranging the indistinct items. The answer is computed by evaluating (10!)/(9!).
##### Problem 2:
Since the house numbers are 7 digits long, there are 7 slots. There are 5 stencils so there are 5 choices for each slot. Stencils are reusable so each of the digits can appear in each slot. The answer is computed by evaluating $$5^7$$.
##### Problem 3:
Starting with a single cell, after d divisions there will be 2^d cells. For example, if the cell divides 3 times, there will be 2^3 cells. If you start with 5 cells instead of 1, there will be 5 times as many cells after 3 divisions or 5 * 2^3.

In general, if you start with n cells there are n*2^d cells after d divisions.

The cells divide every 2 hours so there are 12 divisions per day. They divide for 10 days so there are a total of 120 divisions. The answer is computed by evaluating 8*2^120.
##### Problem 4:
Solve a simpler problem to understand this problem. Assume the building is only 4 stories high. That means he has to climb 3 stories to reach the top. If the building was 5 stories, he would have to climb 4 stories.

In general, the number of stories he climbs is 1 less than the the story number he climbs to. To climb to the 13th story, he climbs 12 stories. When he reaches his goal, the 21st story, he will have climbed 20 stories. The answer then is the reduced form of 12/20 or 3/5.

##### Problem 5:
Work backwards. Since she spent 1/3 of her money at the BBQ, she still had 2/3 of the money left over. The first question then is $10 is 2/3 of what or $10 = 2/3 x

Solve for x by multiplying both sides by the reciprocal of 2/3 or 3/2. That means she had $15 before she entered Nob Hill. She spent 1/6 of her money at Nob Hill so the$15 is 5/6 of the money she had when she entered Nob Hill or

$15 = 5/6 x Solve for x by mulitplying both sides by the reciprocal of 5/6 or 6/5. That means she had$18 when she entered Nob Hill so she spent $18-$15 at Nob Hill or \$3.