Arranging the Letters of the word TRIANGLE to form words that start with T, end with R, start with T and end with R

Problem 1

The letters of the word TRIANGLE are arranged at random. Find the probability and odds that the word so formed (i) starts with T (ii) ends with R (iii) starts with T and ends with R.
Ans :
1
8
,
1
8
,
1
56

Solution

In the word TRIANGLE

Number of Letters/Characters

= 8

{T, R, I, A, N, G, L, E}

⇒ nL = 8

Experiment :

Forming a word using the 8 letters

Total Number of Possible Choices

= Number of words that can be formed using the 8 letters

⇒ n = 8P8
= 8!

When there are large factorial values, using the factorial form would reduce the burden of calculations

Let

A : the event of the word formed starting with T

B : the event of the word formed ending with R

C : the event of the word formed starting with T and ending with R

For Event A

In forming words that start with T,

Number of letters fixed, each in its own place

= 1

⇒ nFL = 1

After fixing the specified letters in their respective places

Number of letters remaining

= Total Number of letters − Number of letters fixed in specific places

⇒ nRL = nL − nFL
= 8 − 1
= 7

Number of places remaining to be filled

= Total Number of Places − Number of letters fixed in specific places

⇒ nRP = nP − nFL
= 8 − 1
= 7

Number of Favorable Choices

= Number of words that can be formed using the letters of the word TRIANGLE fixing T in the first place

= Number of ways in which the specified letters can be fixed each in its own place × Number of ways in which the remaining letters can be arranged in the remaining places

= nFLPnFL × nRLPnRP

= 1 × nRLPnRP

= nRLPnRP

⇒ mA = 7P7
= 7!

Probability that the word formed using all the letters of the word TRIANGLE starts with T

⇒ Probability of occurrence of Event A

=
Number of Favorable Choices for the Event
Total Number of Possible Choices for the Experiment
⇒ P(A) =
mA
n
=
7!
8!
=
7!
8 × 7!
=
1
8

Odds

Number of Unfavorable Choices

= Total Number of possible choices − Number of Favorable choices

⇒ mAc = n − mA
= 8! − 7!
= 8 × 7! − 7!
= (8 − 1) × 7!
= 7 × 7!

in favor

Odds in Favor of the word formed starting with T

⇒ Odds in Favor of Event A

= Number of Favorable Choices : Number of Unfavorable Choices

= mA : mAc

= 7! : 7 × 7!

= 1 : 7

against

Odds against the word formed starting with T

⇒ Odds against Event A

= Number of Unfavorable Choices : Number of Favorable Choices

= mAc : mA

= 7 × 7! : 7!

= 7 : 1

For Event B

In forming words that ends with R

Number of letters fixed, each in its own place

= 1

⇒ nFL = 1

After fixing the specified letters in their respective places

Number of letters remaining

= Total Number of letters − Number of letters fixed in specific places

⇒ nRL = nL − nFL
= 8 − 1
= 7

Number of places remaining to be filled

= Total Number of Places − Number of letters fixed in specific places

⇒ nRP = nP − nFL
= 8 − 1
= 7

Number of Favorable Choices

= Number of words that can be formed using the letters of the word TRIANGLE fixing R in the last place

= Number of ways in which the specified letters are fixed each in its own place × Number of ways in which the remaining letters are arranged in the remaining places

= nFLPnFL × nRLPnRP

= 1 × nRLPnRP

= nRLPnRP

⇒ mA = 7P7
= 7!

Probability that the word formed using all the letters of the word TRIANGLE ends with R

⇒ Probability of occurrence of Event B

=
Number of Favorable Choices for the Event
Total Number of Possible Choices for the Experiment
⇒ P(B) =
mB
n
=
7!
8!
=
7!
8 × 7!
=
1
8

For Event C

In forming words that start with T and end with R,

Number of letters fixed, each in its own place

= 2

⇒ nFL = 2

After fixing the specified letters in their respective places

Number of letters remaining

= Total Number of letters − Number of letters fixed in specific places

⇒ nRL = nL − nFL
= 8 − 2
= 6

Number of places remaining to be filled

= Total Number of Places − Number of letters fixed in specific places

⇒ nRP = nP − nFL
= 8 − 2
= 6

Number of Favorable Choices

= Number of words that can be formed using the letters of the word TRIANGLE fixing T in the first place and R in the last place

= Number of ways in which the specified letters are fixed each in its own place × Number of ways in which the remaining letters are arranged in the remaining places

= nFLPnFL × nRLPnRP

= 1 × nRLPnRP

= nRLPnRP

⇒ mA = 6P6
= 6!

Probability that the word formed using all the letters of the word TRIANGLE starts with T and end with R

⇒ Probability of occurrence of Event C

=
Number of Favorable Choices for the Event
Total Number of Possible Choices for the Experiment
⇒ P(C) =
mC
n
=
6!
8!
=
6!
8 × 7 × 6!
=
1
8 × 7
=
1
56

Odds

Number of Unfavorable Choices

= Total Number of possible choices − Number of Favorable choices

⇒ mAc = n − mA
= 8! − 6!
= 8 × 7 × 6! − 7!
= 56 × 6! − 7!
= (56 − 1) × 6!
= 55 × 6!

in favor

Odds in Favor of the word formed starting with T and ending with R

⇒ Odds in Favor of Event C

= Number of Favorable Choices : Number of Unfavorable Choices

= mC : mCc

= 6! : 55 × 6!

= 1 : 55

against

Odds against the word formed starting with T and ending with R

⇒ Odds against Event C

= Number of Unfavorable Choices : Number of Favorable Choices

= mCc : mC

= 55 × 6! : 6!

= 55 : 1