Question 1156615
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Number of divisors of 1050....<br>
(1) Find the prime factorization of 1050: (2^1)(3^1)(5^2)(7^1)
(2) Add one to each exponent and multiply: 2*2*3*2 = 24<br>
The number of divisors of 1050 is 24.<br>
Explanation....<br>
Given the prime factorization of 1050, every divisor of 1050 can contain only prime factors of 2, 3, 5, and/or 7.  When building a factor of 1050 from the prime factorization, there are...
2 choices for the number of factors of 2 (0 or 1)
2 choices for the number of factors of 3 (0 or 1)
3 choices for the number of factors of 5 (0, 1, or 2)
2 choices for the number of factors of 7 (0 or 1)<br>
The total number of ways to build a divisor of 1050 is then 2*2*3*2.<br>
Solve P(n,3) = 24C(n,4)....<br>
{{{P(n,3) = n(n-1)(n-2)}}}
{{{C(n,4) = (n(n-1)(n-2)(n-3))/(4*3*2*1)}}}<br>
{{{n(n-1)(n-2) = 24*((n(n-1)(n-2)(n-3))/24)}}}<br>
Cancel the common factors n, n-1, and n-2:<br>
{{{1 = n-3}}}
{{{n = 4}}}<br>
ANSWER: n=4<br>
CHECK:<br>
P(4,3) = 4*3*2 = 24
24C(4,4) = 24*1 = 24<br>