Where Mindful Learning Begins
Where Mindful Learning Begins
Free, step-by-step Class 11 Maths NCERT Solutions for Chapter 6 Ex 6.2 — all 5 questions solved, covering factorial notation: evaluating factorials, simplifying factorial ratios, and solving equations built from them.
This short exercise is really about one skill — using n!=n\times(n-1)! to peel off just enough of a factorial to cancel with the denominator, instead of expanding the whole product and dividing. Every question here becomes routine once that cancellation habit is in place, and it is exactly the shortcut nPr and nCr rely on in the exercises ahead.
8!=1\times2\times3\times4\times5\times6\times7\times8=40320
4!=1\times2\times3\times4=24 \qquad 3!=1\times2\times3=6
4!-3!=24-6=18
Evaluating the left-hand side:
3!+4!=6+24=30
Evaluating the right-hand side:
7!=1\times2\times3\times4\times5\times6\times7=5040
Since 30\ne5040, the two sides are not equal.
Write 8! in terms of 6! so it cancels with the denominator:
8!=8\times7\times6!
Substituting:
\dfrac{8!}{6!\times2!}=\dfrac{8\times7\times6!}{6!\times2!}=\dfrac{8\times7}{2!}=\dfrac{56}{2}=28
Write 7! and 8! in terms of 6! so every term shares the same factorial:
7!=7\times6! \qquad 8!=8\times7\times6!
Substituting into the left-hand side:
\dfrac{1}{6!}+\dfrac{1}{7\times6!}=\dfrac{7}{7\times6!}+\dfrac{1}{7\times6!}=\dfrac{8}{7\times6!}
So the equation becomes:
\dfrac{8}{7\times6!}=\dfrac{x}{8\times7\times6!}
Multiplying both sides by 8\times7\times6!:
x=8\times8=64
\dfrac{6!}{(6-2)!}=\dfrac{6!}{4!}=\dfrac{6\times5\times4!}{4!}=6\times5=30
\dfrac{9!}{(9-5)!}=\dfrac{9!}{4!}=\dfrac{9\times8\times7\times6\times5\times4!}{4!}=9\times8\times7\times6\times5=15120
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