Given:
A committee of 6 members is to be chose from the 100 members of the U.S. senate.
To find:
The number of ways to form a committee.
Solution:
We have,
Total number of members = 100
Number of members needs to selected of committee = 6
Number of ways to select r items from total n items is
[tex]^nC_r=\dfrac{n!}{r!(n-r)!}[/tex]
Number of ways to select 6 members from total 100 members is
[tex]^{100}C_6=\dfrac{100!}{6!(100-6)!}[/tex]
[tex]^{100}C_6=\dfrac{100\times 99\times 98\times 97\times 96\times 95\times 94!}{6\times 5\times 4\times 3\times 2\times 1\times 94!}[/tex]
[tex]^{100}C_6=\dfrac{100\times 99\times 98\times 97\times 96\times 95}{6\times 5\times 4\times 3\times 2\times 1}[/tex]
[tex]^{100}C_6=1192052400[/tex]
Therefore, the total number of ways to form a committee is [tex]^{100}C_6[/tex], i.e., equal to [tex]1192052400[/tex].