**Question 11. If P(n, 5) : P(n, 3) = 2 : 1, find n.**

**Solution:**

Given:

P(n, 5) : P(n, 3) = 2 : 1

After applying the formula,

P (n, r) =

P (n, 5) =

P (n, 3) =

So, from the question,

After substituting the values in above expression we will get,

(n – 3)(n – 4) = 2

n

^{2}– 3n – 4n + 12 = 2n

^{2}– 7n + 12 – 2 = 0n

^{2}– 7n + 10 = 0n

^{2}– 5n – 2n + 10 = 0n (n – 5) – 2(n – 5) = 0

(n – 5) (n – 2) = 0

n = 5 or 2

For, P (n, r): n ≥ r

∴ n = 5 [for, P (n, 5)]

**Question 12. Prove that:**

**1. P (1, 1) + 2. P (2, 2) + 3 . P (3, 3) + … + n . P(n, n) = P(n + 1, n + 1) – 1.**

**Solution:**

By using the formula,

P (n, r) =

P (n, n) =

=

= n! [Since, 0! = 1]

Consider LHS:

= 1. P(1, 1) + 2. P(2, 2) + 3. P(3, 3) + … + n . P(n, n)

= 1.1! + 2.2! + 3.3! +………+ n.n! [Since, P(n, n) = n!]

= (2! – 1!) + (3! – 2!) + (4! – 3!) + ……… + (n! – (n – 1)!) + ((n+1)! – n!)

= 2! – 1! + 3! – 2! + 4! – 3! + ……… + n! – (n – 1)! + (n+1)! – n!

= (n + 1)! – 1!

= (n + 1)! – 1 [Since, P (n, n) = n!]

= P(n+1, n+1) – 1

= RHS

Hence Proved.

**Question 13. If P(15, r – 1) : P(16, r – 2) = 3 : 4, find r.**

**Solution:**

Given:

P(15, r – 1) : P(16, r – 2) = 3 : 4

After applying the formula,

P (n, r) =

P (15, r – 1) =

=

P (16, r – 2) =

=

So, from the question,

After substituting the values in above expression we will get,

(18 – r) (17 – r) = 12

306 – 18r – 17r + r

^{2}= 12306 – 12 – 35r + r

^{2}= 0r

^{2}– 35r + 294 = 0r

^{2}– 21r – 14r + 294 = 0r(r – 21) – 14(r – 21) = 0

(r – 14) (r – 21) = 0

r = 14 or 21

For, P(n, r): r ≤ n

∴ r = 14 [for, P(15, r – 1)]

**Question 14. **^{n+5}P_{n}+1 = 11(n – 1)/2 ^{n+3}P_{n}, find n.

^{n+5}P

_{n}+1 = 11(n – 1)/2

^{n+3}P

_{n}, find n.

**Solution:**

Given:

^{n+5}P_{n+1}= 11(n – 1)/2^{n+3}P_{n}P (n +5, n + 1) = 11(n – 1)/2 P(n + 3, n)

By using the formula,

P (n, r) =

P(n + 5, n=1) =

P(n + 3, n) =

So, from the question

P(n + 5, n + 1) = 11(n -1)/2P(n + 3, n)

After substituting the values in above expression we get,

(n + 5) (n + 4) = 22 (n – 1)

n

^{2}+ 4n + 5n + 20 = 22n – 22n

^{2}+ 9n + 20 – 22n + 22 = 0n

^{2}– 13n + 42 = 0n

^{2}– 6n – 7n + 42 = 0n(n – 6) – 7(n – 6) = 0

(n – 7) (n – 6) = 0

n = 7 or 6

∴ The value of n can either be 6 or 7.

**Question 15. In how many ways can five children stand in a queue?**

**Solution:**

Number of arrangements of ‘n’ things taken all at a time = P (n, n)

Hence,

After applying the formula,

P (n, r) =

The total number of ways in which five children can stand in a queue = the number of arrangements of 5 things taken all at a time = P (5, 5)

Thus,

P (5, 5) =

=

= 5! [Since, 0! = 1]

= 5 × 4 × 3 × 2 × 1

= 120

Therefore, Number of ways in which five children can stand in a queue are 120.

**Question 16. From among the 36 teachers in a school, one principal and one vice-principal are to be appointed. In how many ways can this be done?**

**Solution:**

Given:

The total number of teachers in a school = 36

As we know that, number of arrangements of n things taken r at a time = P(n, r)

After applying the formula,

P (n, r) =

∴ The total number of ways in which this can be done = the number of arrangements of 36 things taken 2 at a time = P(36, 2)

P (36, 2) =

=

=

= 36 × 35

= 1260

Hence, Number of ways in which one principal and one vice-principal are to be appointed out of total 36 teachers in school are 1260.

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