Question

Which of the following is/are true?
(i)\[{5^6} - {}^5{C_1}{4^6} + {}^5{C_2}{3^6} - {}^5{C_3}{2^6} + {}^5{C_4}{1^6} = {}^6{C_2}\left| \!{\underline {\, 5 \,}} \right. \]
(ii)\[{6^5} - {}^6{C_1}{5^5} + {}^6{C_2}{4^5} - {}^6{C_3}{3^5} + {}^6{C_4}{2^5} - {}^6{C_1}{1^5} = 0\]
(iii)\[{6^6} - {}^6{C_1}{5^6} + {}^6{C_2}{4^6} - {}^6{C_3}{3^6} + {}^6{C_4}{2^6} - {}^6{C_5}{1^6} = 720\]
(iv)\[{6^5} - {}^6{C_1}{5^5} + {}^6{C_2}{4^5} - {}^6{C_3}{3^5} + {}^6{C_4}{2^5} - {}^6{C_5}{1^5} = {}^5{C_2}\left| \!{\underline {\, 6 \,}} \right. \]

Answer 1

Hint: We can separately solve the left hand side and right hand side. You can use the formulas to derive it and show which statements are true or false.

Formula used:
Here, we used the formula \[^n{C_r} = \dfrac{{n!}}{{\left( {n - r} \right)!r!}}\] , \[n! = n \times \left( {n - 1} \right) \times .... \times 1\] .

Complete step-by-step answer:
(i)\[{5^6} - {}^5{C_1}{4^6} + {}^5{C_2}{3^6} - {}^5{C_3}{2^6} + {}^5{C_4}{1^6} = {}^6{C_2}\left| \!{\underline {\, 5 \,}} \right. \]
We will firstly solve all the combinations present in the statement using the combination formula as shown above.
\[^5{C_1} = \dfrac{{5!}}{{\left( {5 - 1} \right)!1!}}\]
On solving further we get,
\[^5{C_1} = \dfrac{{5!}}{{4!1!}}\]
Solving all the factorial’s factorial formula as shown above.
$\Rightarrow$ \[^5{C_1} = \dfrac{{5 \times 4!}}{{4!}}\]
Cancelling out 4! From both numerator and denominator:
So, we get \[^5{C_1} = 5\]
$\Rightarrow$ \[^5{C_2} = \dfrac{{5!}}{{\left( {5 - 2} \right)!2!}}\]
On solving further we get,
\[^5{C_1} = \dfrac{{5!}}{{3!2!}}\]
Solving all the factorials using factorial formula as shown above.
$\Rightarrow$ \[^5{C_1} = \dfrac{{5 \times 4 \times 3!}}{{3! \times 2}}\]
Cancelling out 3! From both numerator and denominator:
$\Rightarrow$ \[^5{C_1} = \dfrac{{20}}{2}\]
So, we get \[^5{C_1} = 10\]
\[^5{C_3} = \dfrac{{5!}}{{\left( {5 - 3} \right)!3!}}\]
On solving further we get,
$\Rightarrow$ \[^5{C_3} = \dfrac{{5!}}{{2!3!}}\]
Solving all the factorials using the factorial formula as shown above.
$\Rightarrow$ \[^5{C_{_3}} = \dfrac{{5 \times 4 \times 3!}}{{2 \times 3!}}\]
Cancelling out 3! From both numerator and denominator:
$\Rightarrow$ \[^5{C_3} = \dfrac{{20}}{2}\]
So, we get \[^5{C_3} = 10\]
\[^5{C_4} = \dfrac{{5!}}{{\left( {5 - 4} \right)!4!}}\]
On solving further we get,
$\Rightarrow$ \[^5{C_4} = \dfrac{{5!}}{{1!4!}}\]
Solving all the factorial’s factorial formula as shown above.
$\Rightarrow$ \[^5{C_4} = \dfrac{{5 \times 4!}}{{4!}}\]
Cancelling out 4! From both numerator and denominator:
So, we get \[^5{C_4} = 5\]
\[^6{C_2} = \dfrac{{6!}}{{\left( {6 - 2} \right)!2!}}\]
On solving further we get,
$\Rightarrow$ \[^6{C_2} = \dfrac{{6!}}{{4!2!}}\]
Solving all the factorial’s factorial formula as shown above.
\[^6{C_2} = \dfrac{{6 \times 5 \times 4!}}{{4!2!}}\]
Cancelling out 4! From both numerator and denominator:
$\Rightarrow$ \[^6{C_2} = \dfrac{{30}}{2}\]
So, we get \[^6{C_2} = 15\]
Putting all the combination values in L.H.S \[ \Rightarrow {5^6} - {}^5{C_1}{4^6} + {}^5{C_2}{3^6} - {}^5{C_3}{2^6} + {}^5{C_4}{1^6}\]
We get,
\[{5^6} - 5 \times {4^6} + 10 \times {3^6} - 10 \times {2^6} + 5 \times {1^6}\]
\[ \Rightarrow 5\left( {{5^5} - {4^6} + 2 \times {3^6} - 2 \times {2^6} + {1^6}} \right)\]
On simplifying we get,
 \[ \Rightarrow 5\left( {3125 - 4096 + 2 \times 729 - 2 \times 64 + 1} \right)\]
\[ \Rightarrow 5\left( {3125 - 4096 + 1458 - 128 + 1} \right)\]
 \[ \Rightarrow 5\left( {4584 - 4224} \right)\]
 \[ \Rightarrow 5\left( {360} \right)\]
We get, \[{5^6} - {}^5{C_1}{4^6} + {}^5{C_2}{3^6} - {}^5{C_3}{2^6} + {}^5{C_4}{1^6} = 1800\]
Now, we will calculate \[ \Rightarrow {}^6{C_2}\left| \!{\underline {\,
  5 \,}} \right. \]
 Putting all the values in above, \[ \Rightarrow 15 \times 120\]
We get, \[{}^6{C_2}\left| \!{\underline {\,
  5 \,}} \right. = 1800\]
Hence, L.H.S=R.H.S
Therefore, \[{5^6} - {}^5{C_1}{4^6} + {}^5{C_2}{3^6} - {}^5{C_3}{2^6} + {}^5{C_4}{1^6} = {}^6{C_2}\left| \!{\underline {\,
  5 \,}} \right. \] statement is true.
\[{6^5} - {}^6{C_1}{5^5} + {}^6{C_2}{4^5} - {}^6{C_3}{3^5} + {}^6{C_4}{2^5} - {}^6{C_1}{1^5} = 0\]
Similarly, solve all the combinations as done in part (i)
Solving L.H.S, \[{6^5} - {}^6{C_1}{5^5} + {}^6{C_2}{4^5} - {}^6{C_3}{3^5} + {}^6{C_4}{2^5} - {}^6{C_1}{1^5}\]
By putting all the values of combination:
\[ \Rightarrow {6^5} - 6 \times {5^5} + 15 \times {4^5} - 20 \times {3^5} + 15 \times {2^5} - 6 \times {1^5}\]
\[ \Rightarrow 7776 - 6 \times 3125 + 15 \times 1024 - 20 \times 243 + 15 \times 32 - 6 \times 1\]
By simplifying:
\[ \Rightarrow 7776 - 18750 + 15360 - 4860 + 480 - 6\]
\[ \Rightarrow 23616 - 23616\]
We get, \[{6^5} - {}^6{C_1}{5^5} + {}^6{C_2}{4^5} - {}^6{C_3}{3^5} + {}^6{C_4}{2^5} - {}^6{C_1}{1^5} = 0\] which is equal to R.H.S.
Therefore, \[{6^5} - {}^6{C_1}{5^5} + {}^6{C_2}{4^5} - {}^6{C_3}{3^5} + {}^6{C_4}{2^5} - {}^6{C_1}{1^5} = 0\] statement is true.

\[{6^6} - {}^6{C_1}{5^6} + {}^6{C_2}{4^6} - {}^6{C_3}{3^6} + {}^6{C_4}{2^6} - {}^6{C_5}{1^6} = 720\]
Similarly, solve all the combinations as done in part (i)
 Solving L.H.S, \[{6^6} - {}^6{C_1}{5^6} + {}^6{C_2}{4^6} - {}^6{C_3}{3^6} + {}^6{C_4}{2^6} - {}^6{C_5}{1^6}\]
By putting all the values of combination:
 \[ \Rightarrow {6^6} - 6 \times {5^6} + 15 \times {4^6} - 20 \times {3^6} + 15 \times {2^6} - 6 \times {1^6}\]
\[ \Rightarrow 46656 - 6 \times 15625 + 15 \times 4096 - 20 \times 729 + 15 \times 64 - 6 \times 1\]
 By simplifying:
\[ \Rightarrow 46656 - 93750 + 61440 - 14580 + 960 - 6\]
\[ \Rightarrow 109056 - 108336\]
 We get, \[{6^6} - {}^6{C_1}{5^6} + {}^6{C_2}{4^6} - {}^6{C_3}{3^6} + {}^6{C_4}{2^6} - {}^6{C_5}{1^6} = 720\] which is equal to R.H.S.
Therefore, \[{6^6} - {}^6{C_1}{5^6} + {}^6{C_2}{4^6} - {}^6{C_3}{3^6} + {}^6{C_4}{2^6} - {}^6{C_5}{1^6} = 720\] statement is true.

(iv)\[{6^5} - {}^6{C_1}{5^5} + {}^6{C_2}{4^5} - {}^6{C_3}{3^5} + {}^6{C_4}{2^5} - {}^6{C_5}{1^5} = {}^5{C_2}\left| \!{\underline {\, 6 \,}} \right. \]
  Similarly, solve all the combinations as done in part (i)
Solving L.H.S, \[{6^5} - {}^6{C_1}{5^5} + {}^6{C_2}{4^5} - {}^6{C_3}{3^5} + {}^6{C_4}{2^5} - {}^6{C_5}{1^5}\]
By putting all the values of combination:
\[ \Rightarrow {6^5} - 6 \times {5^5} + 15 \times {4^5} - 20 \times {3^5} + 15 \times {2^5} - 6 \times {1^5}\]
\[ \Rightarrow 7776 - 6 \times 3125 + 15 \times 1024 - 20 \times 243 + 15 \times 32 - 6 \times 1\]
By simplifying:
\[ \Rightarrow 7776 - 18750 + 15360 - 4860 + 480 - 6\]
\[ \Rightarrow 23616 - 23616\]
We get, \[{6^5} - {}^6{C_1}{5^5} + {}^6{C_2}{4^5} - {}^6{C_3}{3^5} + {}^6{C_4}{2^5} - {}^6{C_5}{1^5} = 0\]
On solving R.H.S \[{}^5{C_2}\left| \!{\underline {\,
  6 \,}} \right. \] we get,
\[ \Rightarrow 10 \times 6!\]
\[ \Rightarrow 10 \times 720\]
\[ \Rightarrow 7200\]
Hence, L.H.S \[ \ne \]R.H.S
Therefore, \[{6^5} - {}^6{C_1}{5^5} + {}^6{C_2}{4^5} - {}^6{C_3}{3^5} + {}^6{C_4}{2^5} - {}^6{C_5}{1^5} = {}^5{C_2}\left| \!{\underline {\,
  6 \,}} \right. \] statement is false.

Note: These types of questions are done with the help of combination and factorial formulas. Do the calculation very carefully and make the calculations as simple as you can. Questions like these can be lengthy but you have to follow the same steps most of the cases.