Question

Find the exponent of 10 in \[{}^{75}{{C}_{25}}\].

Answer 1

Hint: To find the exponent of 10 in \[{}^{75}{{C}_{25}}\], we should learn the expansion of \[{}^{n}{{C}_{r}}\], which is equal to \[\dfrac{n!}{r!\left( n-r \right)!}\]. Also, we should know that, power of some positive prime integer ‘m’ which is \[\le n\], in \[n!\] is \[\left[ \dfrac{n}{m} \right]+\left[ \dfrac{n}{{{m}^{2}}} \right]+\left[ \dfrac{n}{{{m}^{3}}} \right]+\left[ \dfrac{n}{{{m}^{4}}} \right]+.....\], where \[\left[ . \right]\] represents the greatest integer number.

Complete step-by-step answer:
We know that \[{}^{75}{{C}_{25}}\] can be expressed using the formula \[{}^{n}{{C}_{r}}=\dfrac{n!}{r!\left( n-r \right)!}\], as \[{}^{75}{{C}_{25}}=\dfrac{75!}{25!\left( 75-25 \right)!}\], where n = 75 and r = 25.
\[\Rightarrow {}^{75}{{C}_{25}}=\dfrac{75!}{25!\left( 50 \right)!}\]
Now, in this question, we have to find the exponent of 10 in \[{}^{75}{{C}_{25}}\], for that we have to find the exponent of 10 in \[75!\], \[25!\] and \[50!\].
As we know, 10 can be represented as \[10=5\times 2\], so the minimum of the exponents of 5 or 2 will be the exponent of 10 in respective numbers.
Now, let us find the exponent of 5 in \[75!\], as we know that power of some positive prime integer ‘m’ which is \[\le n\], in \[n!\] is \[\left[ \dfrac{n}{m} \right]+\left[ \dfrac{n}{{{m}^{2}}} \right]+\left[ \dfrac{n}{{{m}^{3}}} \right]+\left[ \dfrac{n}{{{m}^{4}}} \right]+.....\], where \[\left[ . \right]\] represents the greatest integer number.
So exponent of 5 is\[\left[ \dfrac{75}{5} \right]+\left[ \dfrac{75}{{{5}^{2}}} \right]+\left[ \dfrac{75}{{{5}^{3}}} \right]+.....\]
\[=\left[ \dfrac{75}{5} \right]+\left[ \dfrac{75}{25} \right]+\left[ \dfrac{75}{125} \right]+.....\]
\[=\left[ 15 \right]+\left[ 3 \right]+\left[ \dfrac{3}{5} \right]+.....\]
\[=15+3+0\]
\[=18\]
Now, let us find the exponent of 2 in \[75!\],
So, exponent of 2 is\[\left[ \dfrac{75}{2} \right]+\left[ \dfrac{75}{{{2}^{2}}} \right]+\left[ \dfrac{75}{{{2}^{3}}} \right]+.....\]
\[=\left[ \dfrac{75}{2} \right]+\left[ \dfrac{75}{4} \right]+\left[ \dfrac{75}{8} \right]+\left[ \dfrac{75}{16} \right]+\left[ \dfrac{75}{32} \right]+\left[ \dfrac{75}{64} \right]+.....\]
\[=\left[ 37.5 \right]+\left[ 18.75 \right]+\left[ 9.3 \right]+\left[ 4.6 \right]+\left[ 2.3 \right]+\left[ 1.15 \right]+\left[ 0.57 \right].....\]
\[=37+18+9+4+2+1+0\]
\[=71\]
As the exponent of 5 is smaller than exponent of 2, that is, 18, we can say that exponent of 10 in \[75!\] is 18.
Similarly, we will find the exponent of 10 in \[25!\].
Now, let us find the exponent of 5 in \[25!\],
So, exponent of 5 is\[\left[ \dfrac{25}{5} \right]+\left[ \dfrac{25}{{{5}^{2}}} \right]+\left[ \dfrac{25}{{{5}^{3}}} \right]+.....\]
\[=\left[ \dfrac{25}{5} \right]+\left[ \dfrac{25}{25} \right]+\left[ \dfrac{25}{125} \right]+.....\]
\[=\left[ 5 \right]+\left[ 1 \right]+\left[ \dfrac{1}{5} \right]+.....\]
\[=5+1+0\]
\[=6\]
Now, let us find the exponent of 2 in \[25!\],
So, exponent of 2 is\[\left[ \dfrac{25}{2} \right]+\left[ \dfrac{25}{{{2}^{2}}} \right]+\left[ \dfrac{25}{{{2}^{3}}} \right]+.....\]
\[=\left[ \dfrac{25}{2} \right]+\left[ \dfrac{25}{4} \right]+\left[ \dfrac{25}{8} \right]+\left[ \dfrac{25}{16} \right]+\left[ \dfrac{25}{32} \right]+......\]
\[=\left[ 12.5 \right]+\left[ 6.25 \right]+\left[ 3.12 \right]+\left[ 1.56 \right]+\left[ 0.78 \right]......\]
\[=12+6+3+1+0\]
\[=22\]
As the exponent of 5 is smaller than exponent of 2, that is, 6, we can say that exponent of 10 in \[25!\] is 6.
Similarly, we will find the exponent of 10 in \[50!\].
Now, let us find the exponent of 5 in \[50!\],
So, exponent of 5 is\[\left[ \dfrac{50}{5} \right]+\left[ \dfrac{50}{{{5}^{2}}} \right]+\left[ \dfrac{50}{{{5}^{3}}} \right]+.....\]
\[=\left[ \dfrac{50}{5} \right]+\left[ \dfrac{50}{25} \right]+\left[ \dfrac{50}{125} \right]+.....\]
\[=\left[ 10 \right]+\left[ 2 \right]+\left[ \dfrac{2}{5} \right]+.....\]
\[=10+2+0\]
\[=12\]
Now, let us find the exponent of 2 in \[50!\],
So, exponent of 2 is\[\left[ \dfrac{50}{2} \right]+\left[ \dfrac{50}{{{2}^{2}}} \right]+\left[ \dfrac{50}{{{2}^{3}}} \right]+.....\]
\[=\left[ \dfrac{50}{2} \right]+\left[ \dfrac{50}{4} \right]+\left[ \dfrac{50}{8} \right]+\left[ \dfrac{50}{16} \right]+\left[ \dfrac{50}{32} \right]+\left[ \dfrac{50}{64} \right]+......\]
\[=\left[ 25 \right]+\left[ 12.5 \right]+\left[ 6.25 \right]+\left[ 3.12 \right]+\left[ 1.56 \right]+\left[ 0.78 \right]......\]
\[=25+12+6+3+1+0\]
\[=47\]
As the exponent of 5 is smaller than the exponent of 2, that is, 12, we can say that exponent of 10 in \[50!\] is 12.
Now, to find the exponent of 10 in \[{}^{75}{{C}_{25}}\] which is equal to \[\dfrac{75!}{25!50!}\], we will put the exponents of respective factorials at their respective places, so we get,
Exponent of 10 in \[{}^{75}{{C}_{25}}\] = \[\dfrac{{{10}^{18}}}{{{10}^{6}}\times {{10}^{12}}}\]
\[=\dfrac{{{10}^{18}}}{{{10}^{18}}}\]
\[={{10}^{0}}\]
Hence, the exponent of 10 in \[{}^{75}{{C}_{25}}\] is 0.

Note: The possible mistake one can make is while finding the exponent of 10 in \[75!\], \[25!\] and \[50!\], that is, one can find an exponent of 10 without finding an exponent of 5 and 2 which will lead to the wrong solution. Also, one can mistake by writing \[{{10}^{0}}\] as 1, which can be confusing. Because \[{{10}^{0}}\] means an exponent of 10 in \[{}^{75}{{C}_{25}}\] is 0.