The number of ways in which 7 pictures can be hung from 5 picture nails on a wall is nothing but the number of permutation of 7 things taken 5 at a time
= \({ 7P }_{ 5 }=\cfrac { 7! }{ (7-5)! } =2520\ ways\)

8P₃ = 8 \(\times\) 7 \(\times\) 6 = 336.

  \(\cfrac { 1 }{ 9! } +\cfrac { 1 }{ 10! } =\cfrac { n }{ 11! } \)
\(\cfrac { 1 }{ 9! } +\cfrac { 1 }{ 10\times 9! } =\cfrac { n }{ 11! }\)
\( \cfrac { 1 }{ 9! } \left[ 1+\cfrac { 1 }{ 10 } \right] =\cfrac { n }{ 11! } \)
\( \cfrac { 1 }{ 9! } \times \cfrac { 11 }{ 10 } =\cfrac { n }{ 11! } \)
\(n=\cfrac { 11!\times 11 }{ 9!\times 10 } \)
\(=\cfrac { 11!\times 11 }{ 10! } \)
\( =\cfrac { 11\times 10!\times 11 }{ 10! } \)
\(n=121\)

Let us allot four vacant places (boxes) to represent the four letters. N,O,T,E of the given word NOTE.

The first place can be filled by any one of the 4 letters in 4 ways. Since repetition is not allowed, the number of ways of filling the second vacant place by any of the remaining 3 letters in 3 ways. In the similar way the third box can be filled in 2 ways and that of last box can be filled in 1 way. Hence by multiplication principle of counting, total number of words formed is 4 \(\times\) 3 \(\times\) 2 \(\times\) 1 = 24 words.

Write the denominator into the product of linear factors.
Here x² - 5x + 6 = (x - 2)(x - 3)
Let \(\frac { 7x-1 }{ { x }^{ 2 }-5x+6 } =\frac{A}{(x-2)}+\frac{B}{(x-3)}\)...(1)
Multiplying both the sides of (1) by (x - 2)(x - 3), we get
7x - 1 = A(x - 3) + B(x - 2)....(2)
Put x = 3 in (2),we get
21 – 1 = B (1)
\(\Rightarrow \) B = 20
Put x = 2 in (2),we get
14 - 1 = A(-1)
\(\Rightarrow \) A = -13
Substituting the values of A and B in (1), we get
\(\frac { 7x-1 }{ { x }^{ 2 }-5x+6 } =\frac{-13}{(x-2)}+\frac{20}{(x-3)}\)